Page 1 of 13 Pages


                       SECURITIES AND EXCHANGE COMMISSION
                            WASHINGTON, D.C.  20549
    ----------------------------------------------------------------------
                                  SCHEDULE 13G

           Information Statement pursuant to Rule 13d-1 and 13d-2

                             (AMMENDMENT NO.1)*

                                  HALLIBURTON CO
                  ---------------------------------------------
                                (NAME OF ISSUER)

                                  COMMON STOCK
                  ---------------------------------------------
                         (TITLE OF CLASS OF SECURITIES)
                                   406216101
                  ---------------------------------------------
                                  (CUSIP NUMBER)
    ----------------------------------------------------------------------
    Check the following box if a fee is being paid with this statement :_:
    (A fee is not required only if the filing person: (1) has a previous
    statement on file reporting beneficial ownership of more than five
    percent of the class of securities described in Item 1; and (2) has
    filed no amendment subsequent thereto reporting beneficial ownership
    of five percent or less of such class.) (See Rule 13d-7.)

    * The remainder of this cover page shall be filled out for a reporting
    person`s initial filing on this form with respect to the subject class
    of securities, and for any subsequent amendment containing information
    which would alter the disclosures provided in a prior cover page.

    The information required in the remainder of this cover page shall not
    be deemed to be 'filed' for the purpose of Section 18 of the Securities
    Exchange Act of 1934 ('Act') or otherwise subject to the liabilities
    of that section of the Act but shall be subject to all other provisions
    of the Act (however, see the Notes).

                       (CONTINUED ON FOLLOWING PAGE(S))



    --------------------                          --------------------
    CUSIP NO. 406216101          13G               Page 2 of 13 Pages
    --------------------                          --------------------
    --------------------------------------------------------------------
     1- NAME OF REPORTING PERSON
      AXA Assurances I.A.R.D. Mutuelle

      S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

    -------------------------------------------------------------------
     2- CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP *   (A) [ ]
                                                             (B) [X]
    -------------------------------------------------------------------
     3- SEC USE ONLY


    -------------------------------------------------------------------
     4- CITIZENSHIP OR PLACE OF ORGANIZATION

      France
    -------------------------------------------------------------------
    NUMBER OF SHARES BENEFICIALLY OWNED
    AS OF September 30, 1998 BY EACH REPORTING PERSON WITH
                -----------------------------------------------
                5- SOLE VOTING POWER
                     13,039,010
                -----------------------------------------------
                6- SHARED VOTING POWER
                     12,363,370
                -----------------------------------------------
                7- SOLE DISPOSITIVE POWER
                     29,504,460
                -----------------------------------------------
                8- SHARED DISPOSITIVE POWER
                     30,341
    --------------------------------------------------------------------
     9- AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
                29,564,860
      (Not to be construed as an admission of beneficial ownership)
    -------------------------------------------------------------------
    10- CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN
      SHARES *                                            |------|
                                                          |------|
    -------------------------------------------------------------------
    11- PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9

                         11.2%
    -------------------------------------------------------------------
    12- TYPE OF REPORTING PERSON *

                                   IC
    --------------------------------------------------------------------

                    * SEE INSTRUCTIONS BEFORE FILLING OUT!



    --------------------                          --------------------
    CUSIP NO. 406216101          13G               Page 3 of 13 Pages
    --------------------                          --------------------
    --------------------------------------------------------------------
     1- NAME OF REPORTING PERSON
      AXA Assurances Vie Mutuelle

      S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

    -------------------------------------------------------------------
     2- CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP *   (A) [ ]
                                                             (B) [X]
    -------------------------------------------------------------------
     3- SEC USE ONLY


    -------------------------------------------------------------------
     4- CITIZENSHIP OR PLACE OF ORGANIZATION

      France
    --------------------------------------------------------------------
    NUMBER OF SHARES BENEFICIALLY OWNED
    AS OF September 30, 1998 BY EACH REPORTING PERSON WITH
                -----------------------------------------------
                5- SOLE VOTING POWER
                     13,039,010
                -----------------------------------------------
                6- SHARED VOTING POWER
                     12,363,370
                -----------------------------------------------
                7- SOLE DISPOSITIVE POWER
                     29,504,460
                -----------------------------------------------
                8- SHARED DISPOSITIVE POWER
                     30,341
    --------------------------------------------------------------------
     9- AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
                29,564,860
      (Not to be construed as an admission of beneficial ownership)
    -------------------------------------------------------------------
    10- CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN
      SHARES *                                            |------|
                                                          |------|
    -------------------------------------------------------------------
    11- PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9

                         11.2%
    -------------------------------------------------------------------
    12- TYPE OF REPORTING PERSON *

                                   IC
    --------------------------------------------------------------------

                    * SEE INSTRUCTIONS BEFORE FILLING OUT!



    --------------------                          --------------------
    CUSIP NO. 406216101          13G               Page 4 of 13 Pages
    --------------------                          --------------------
    --------------------------------------------------------------------
     1- NAME OF REPORTING PERSON
      Alpha Assurances Vie Mutuelle

      S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

    -------------------------------------------------------------------
     2- CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP *   (A) [ ]
                                                             (B) [X]
    -------------------------------------------------------------------
     3- SEC USE ONLY


    -------------------------------------------------------------------
     4- CITIZENSHIP OR PLACE OF ORGANIZATION

      France
    --------------------------------------------------------------------
    NUMBER OF SHARES BENEFICIALLY OWNED
    AS OF September 30, 1998 BY EACH REPORTING PERSON WITH
                -----------------------------------------------
                5- SOLE VOTING POWER
                     13,039,010
                -----------------------------------------------
                6- SHARED VOTING POWER
                     12,363,370
                -----------------------------------------------
                7- SOLE DISPOSITIVE POWER
                     29,504,460
                -----------------------------------------------
                8- SHARED DISPOSITIVE POWER
                     30,341
    --------------------------------------------------------------------
     9- AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
                29,564,860
      (Not to be construed as an admission of beneficial ownership)
    -------------------------------------------------------------------
    10- CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN
      SHARES *                                            |------|
                                                          |------|
    -------------------------------------------------------------------
    11- PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9

                        11.2%
    -------------------------------------------------------------------
    12- TYPE OF REPORTING PERSON *

                                   IC
    --------------------------------------------------------------------

                    * SEE INSTRUCTIONS BEFORE FILLING OUT!



    --------------------                          --------------------
    CUSIP NO. 406216101         13G                Page 5 of 13 Pages
    --------------------                          --------------------
    --------------------------------------------------------------------
     1- NAME OF REPORTING PERSON
      AXA Courtage Assurance Mutuelle

      S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

    -------------------------------------------------------------------
     2- CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP *   (A) [ ]
                                                             (B) [X]
    -------------------------------------------------------------------
     3- SEC USE ONLY


    -------------------------------------------------------------------
     4- CITIZENSHIP OR PLACE OF ORGANIZATION

      France
    --------------------------------------------------------------------
    NUMBER OF SHARES BENEFICIALLY OWNED
    AS OF September 30, 1998 BY EACH REPORTING PERSON WITH
                -----------------------------------------------
                5- SOLE VOTING POWER
                     13,039,010
                -----------------------------------------------
                6- SHARED VOTING POWER
                     12,363,370
                -----------------------------------------------
                7- SOLE DISPOSITIVE POWER
                     29,504,460
                -----------------------------------------------
                8- SHARED DISPOSITIVE POWER
                     30,341
    --------------------------------------------------------------------
     9- AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
                29,564,860
      (Not to be construed as an admission of beneficial ownership)
    -------------------------------------------------------------------
    10- CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN
      SHARES *                                            |------|
                                                          |------|
    -------------------------------------------------------------------
    11- PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9

                     11.2%
    -------------------------------------------------------------------
    12- TYPE OF REPORTING PERSON *

                                   IC
    --------------------------------------------------------------------

                    * SEE INSTRUCTIONS BEFORE FILLING OUT!



    --------------------                          --------------------
    CUSIP NO. 406216101          13G               Page 6 of 13 Pages
    --------------------                          --------------------
    --------------------------------------------------------------------
     1- NAME OF REPORTING PERSON
      AXA-UAP

      S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON

    -------------------------------------------------------------------
     2- CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP *   (A) [ ]
                                                             (B) [ ]
    -------------------------------------------------------------------
     3- SEC USE ONLY


    -------------------------------------------------------------------
     4- CITIZENSHIP OR PLACE OF ORGANIZATION

      France
    --------------------------------------------------------------------
    NUMBER OF SHARES BENEFICIALLY OWNED
    AS OF September 30, 1998 BY EACH REPORTING PERSON WITH
                -----------------------------------------------
                5- SOLE VOTING POWER
                     13,039,010
                -----------------------------------------------
                6- SHARED VOTING POWER
                     12,363,370
                -----------------------------------------------
                7- SOLE DISPOSITIVE POWER
                     29,504,460
                -----------------------------------------------
                8- SHARED DISPOSITIVE POWER
                     30,341
    --------------------------------------------------------------------
     9- AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
                29,564,860
      (Not to be construed as an admission of beneficial ownership)
    -------------------------------------------------------------------
    10- CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN
      SHARES *                                            |------|
                                                          |------|
    -------------------------------------------------------------------
    11- PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9

                             11.2%
    -------------------------------------------------------------------
    12- TYPE OF REPORTING PERSON *

                                   HC
    --------------------------------------------------------------------

                    * SEE INSTRUCTIONS BEFORE FILLING OUT!



    --------------------                          --------------------
    CUSIP NO. 406216101          13G               Page 7 of 13 Pages
    --------------------                          --------------------
    --------------------------------------------------------------------
     1- NAME OF REPORTING PERSON
      THE EQUITABLE COMPANIES INCORPORATED

      S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON
      13-3623351
    -------------------------------------------------------------------
     2- CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP *   (A) [ ]
                                                             (B) [ ]
    -------------------------------------------------------------------
     3- SEC USE ONLY


    -------------------------------------------------------------------
     4- CITIZENSHIP OR PLACE OF ORGANIZATION

      State of Delaware
    --------------------------------------------------------------------
    NUMBER OF SHARES BENEFICIALLY OWNED
    AS OF September 30, 1998 BY EACH REPORTING PERSON WITH
                -----------------------------------------------
                5- SOLE VOTING POWER
                     13,032,913
                -----------------------------------------------
                6- SHARED VOTING POWER
                     12,363,370
                -----------------------------------------------
                7- SOLE DISPOSITIVE POWER
                     29,498,364
                -----------------------------------------------
                8- SHARED DISPOSITIVE POWER
                     30,341
    --------------------------------------------------------------------
     9- AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
                29,558,755

    -------------------------------------------------------------------
    10- CHECK BOX IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES CERTAIN
      SHARES *                                            |------|
                                                          |------|
    -------------------------------------------------------------------
    11- PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW 9

                 11.2%
    -------------------------------------------------------------------
    12- TYPE OF REPORTING PERSON *

                                  HC
    --------------------------------------------------------------------

                    * SEE INSTRUCTIONS BEFORE FILLING OUT!



    Item 1(a) Name of Issuer:                         Page 8 of 13 Pages
              --------------
              HALLIBURTON CO

    Item 1(b) Address of Issuer's Principal Executive Offices:
              -----------------------------------------------
              3600 Lincoln Plaza
              Dallas, TX 75201


    Item 2(a) Name of Person Filing:
              ---------------------
              Alpha Assurances Vie Mutuelle,
              AXA Assurances I.A.R.D Mutuelle,
              AXA Assurances Vie Mutuelle,
              and AXA Courtage Assurance Mutuelle,
              as a group (collectively, the 'Mutuelles AXA').

              AXA-UAP

              The Equitable Companies Incorporated
                (the 'Equitable Companies')

                (Please contact Patrick Meehan at (212) 641-8234
                            with any questions.)

    Item 2(b) Address of Principal Business Office:
              ------------------------------------
              Alpha Assurances Vie Mutuelle
              100-101 Terrasse Boieldieu
              92042 Paris La Defense France

              AXA Assurances I.A.R.D Mutuelle and
              AXA Assurances Vie Mutuelle
              21, rue de Chateaudun
              75009 Paris France

              AXA Courtage Assurance Mutuelle
              26, rue Louis le Grand
              75002 Paris France

              AXA-UAP
              9 Place Vendome
              75001 Paris France

              The Equitable Companies Incorporated
              1290 Avenue of the Americas
              New York, New York 10104




                                                      Page  9 of 13 Pages


    Item 2(c) Citizenship:
              -----------
              Mutuelles AXA and AXA-UAP France
              Equitable Companies - Delaware

    Item 2(d) Title of Class of Securities:
              ----------------------------
              Common Stock

    Item 2(e) CUSIP Number:
              ------------
              406216101

    Item 3.   Type of Reporting Person:
              ------------------------
              Equitable Companies as a parent holding company,
              in accordance with 240.13d-1 (b)(ii)(G).

              The Mutuelles AXA, as a group, acting as a parent
              holding company.

              AXA-UAP as a parent holding company.




                                                    Page 10 of 13 Pages
    Item 4. Ownership as of September 30, 1998:
            ------------------------------
    (a) Amount Beneficially Owned:
        -------------------------
       29,564,860 shares of common stock beneficially owned including:
No. of Shares --------------------- The Mutuelles AXA, as a group 0 AXA-UAP 0 AXA-UAP Entity or Entities: National Mutual Funds Management (Australia) Common Stock 5,800 5,800 Sun Life & Provincial Holdings PLC (U.K.) Common Stock 300 300 ---------- acquired solely for investment purposes: (Each of the Mutuelles AXA, as a group, and AXA-UAP expressly declares that the filing of this Schedule 13G shall not be construed as an admission that it is, for purposes of Section 13(d) of the Exchange Act, the beneficial owner of any securities covered by this Schedule 13G). The Equitable Companies Incorporated 0 Subsidiaries: ------------ Alliance Capital Management L.P. acquired solely for investment purposes on behalf of client discretionary investment advisory accounts: Common Stock 27,406,881 27,406,881 ---------- Donaldson, Lufkin & Jenrette Securities Corporation held for investment purposes: Common Stock 638,779 Shares which may be acquired/(disposed of) upon exercise of options -39,800 598,979 ---------- The Equitable Life Assurance Society of the United States acquired solely for investment purposes: Common Stock 1,169,900 1,169,900 ---------- Wood, Struthers & Winthrop Management Corporation acquired solely for investment purposes on behalf of client discretionary investment advisory accounts: Common Stock 382,995 382,995 ---------- ----------- Total 29,564,860 =========== (Each of the above subsidiaries of The Equitable operates under independent management and makes independent decisions.) (b) Percent of Class: 11.2% ---------------- ===========
ITEM 4. Ownership as of 9/30/98(CONT.) Page 11 of 13 Pages (c) Deemed Voting Power and Disposition Power: -----------------------------------------
(i) (ii) (iii) (iv) Deemed Deemed Deemed Deemed to have to have to have to have Sole Power Shared Power Sole Power Shared Power to Vote to Vote to Dispose to Dispose or to or to or to or to Direct Direct Direct the Direct the the Vote the Vote Disposition Disposition ------------ ------------ ------------ ------------ The Mutuelles AXA, as a group 0 0 0 0 AXA-UAP 0 0 0 0 AXA-UAP Entity or Entities: ------------------ National Mutual Funds Management (Australia) 5,800 0 5,800 0 Sun Life & Provincial Holdings PLC (U.K.) 300 0 300 0 The Equitable Companies Incorporated 0 0 0 0 Subsidiaries: ------------ Alliance Capital Management L.P. 11,442,825 11,904,260 27,406,721 160 Donaldson, Lufkin & Jenrette Securities Corporation 585,298 0 568,798 30,181 The Equitable Life Assurance Society of the United States 836,500 333,400 1,169,900 0 Wood, Struthers & Winthrop Management Corporation 168,290 125,710 352,945 0 ------------ ------------ ------------ ------------ TOTAL 13,039,010 12,363,370 29,504,460 30,341 ============ ============ ============ ============ (Each of the above subsidiaries of the Equitable Companies operates under independent management and makes independent voting and investment decisions).
Page 12 of 13 Pages Item 5. Ownership of Five Percent or Less of a Class: --------------------------------------------- If this statement is being filed to report the fact that as of the date hereof the reporting person has ceased to be the beneficial owner of more than five percent of the class of securities, check the following. ( ) Item 6. Ownership of More than Five Percent on behalf of Another Person. N/A ---------------------------------------------------------------- Item 7. Identification and Classification of the Subsidiary which Acquired ------------------------------------------------------------------ the Security Being Reporting on by the Parent Holding Company: ------------------------------------------------------------- This Schedule 13G is being filed by Equitable Companies; AXA-UAP, which beneficially owns a majority interest in Equitable Companies; and the Mutuelles AXA, which as a group control AXA-UAP: (X) in the Mutuelles AXAs' capacity, as a group, acting as a parent holding company with respect to the holdings of the following AXA-UAP entity or entities; (X) in AXA-UAP's capacity as a parent holding company with respect to the holdings of the following AXA-UAP entity or entities: National Mutual Funds Management (Australia) Sun Life & Provincial Holdings PLC (U.K.) (X) in EQUITABLE COMPANIES capacity as a parent holding company with respect to the holdings of its following subsidiaries: (X) Alliance Capital Management L.P. (13-3434400), an investment adviser registered under Section 203 of the Investment Advisers Act of 1940. (X) Donaldson, Lufkin & Jenrette Securities Corporation (13-2741729), a broker-dealer registered under Section 15 of the Securities and Exchange Act of 1934 and an investment adviser registered under Section 203 of the Investment Advisers Act of 1940. (X) The Equitable Life Assurance Society of the United States (13-5570651), an insurance company, a broker-dealer registered under Section 15 of the Securities and Exchange Actof 1934 and an investment adviser registered under Section 203 of the Investment Advisers Act of 1940. (X) Wood, Struthers & Winthrop Management Corporation (13-2774791), an investment adviser registered under Section 203 of the Investment Advisers Act of 1940. Page 13 of 13 Pages Item 8. Identification and Classification of Members of the Group. N/A --------------------------------------------------------- Item 9. Notice of Dissolution of Group: N/A ------------------------------ Item 10. Certification: ------------- By signing below I certify that to the best of my knowledge and belief, the securities referred to above were acquired in the ordinary course of business and were not acquired for the purpose of and do not have the effect of changing or influencing the control of the issuer of such securities and were not acquired in connection with or as a participant in any transaction having such purposes or effect. Signature --------- After reasonable inquiry and to the best of my knowledge and belief, I certify that the information set forth in this statement is true, complete and correct. Date: October 13, 1998 THE EQUITABLE COMPANIES INCORPORATED* /s/ Alvin H. Fenichel -------------------------------------------- Alvin H. Fenichel Senior Vice President and Controller *Pursuant to the Joint Filing Agreement with respect to Schedule 13G attached hereto as Exhibit I, among The Equitable Companies Incorporated, Alpha Assurances Vie Mutuelle, AXA Assurances I.A.R.D Mutuelle, AXA Assurances Vie Mutuelle, AXA Courtage Assurance Mutuelle, and AXA-UAP, this statement Schedule 13G is filed on behalf of each of them.
                                                                 EXHIBIT I



                                   JOINT FILING AGREEMENT
                                   ----------------------




                       Each of the undersigned hereby agrees that the
               Schedule 13G filed herewith is filed jointly, pursuant to
               Rule 13d-1(f)(1) of the Securities Exchange Act of 1934,
               as amended on behalf of each of them.


               Dated: October 13, 1998



               The Equitable Companies Incorporated



               BY:  /s/ Alvin H. Fenichel
                  -------------------------
                  Alvin H. Fenichel
                  Senior Vice President
                  and Controller



               AXA Assurances I.A.R.D. Mutuelle; AXA Assurances Vie Mutuelle;
               Alpha Assurances Vie Mutuelle; AXA Courtage Assurance Mutuelle,
               as a group, and AXA-UAP

               Signed on behalf of each of the above entities



               BY:      /s/ Alvin H. Fenichel
                  -----------------------------------------
                  Alvin H. Fenichel
                  Attorney-in-Fact
                  (Executed pursuant to Powers of Attorney)