UNITED STATES SECURITIES AND EXCHANGE COMMISSION Washington, D.C. 20549 SCHEDULE 13G Under the Securities Exchange Act of 1934* Halliburton Company - -------------------------------------------------------------------------------- (Name of Issuer) Common Stock - -------------------------------------------------------------------------------- (Title of Class of Securities) 406216101 - -------------------------------------------------------------------------------- (CUSIP Number) December 31, 1998 - -------------------------------------------------------------------------------- (Date of Event Which Requires Filing of this Statement) Check appropriate box to designate the rule pursuant to which this Schedule is filed: [X] Rule 13d-1(b) [ ] Rule 13d-1(c) [ ] Rule 13d-1(d) *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 disclosures provided in a prior cover page. The information required on 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). Page 1 of 4 Pages

CUSIP No. 406216101 13G - --------------------- 1. NAME OF REPORTING PERSON S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON Barrow, Hanley, Mewhinney & Strauss, Inc. 752403190 - ---------------------------------------------------------------------------------- 2. (a) [ ] CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP (See Instructions) (b) [ ] - ---------------------------------------------------------------------------------- 3. SEC USE ONLY - ---------------------------------------------------------------------------------- 4. CITIZENSHIP OR PLACE OF ORGANIZATION A Nevada corporation - ---------------------------------------------------------------------------------- 5. SOLE VOTING POWER NUMBER OF 4,169,700 shares SHARES BENEFICIALLY 6. SHARED VOTING POWER OWNED BY 17,876,260 shares EACH REPORTING 7. SOLE DISPOSITIVE POWER PERSON 22,045,960 shares WITH 8. SHARED DISPOSITIVE POWER ---- - ---------------------------------------------------------------------------------- 9. AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON 22,045,960 shares - ---------------------------------------------------------------------------------- 10. CHECK IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES [ ] CERTAIN SHARES (See Instructions) - ---------------------------------------------------------------------------------- 11. PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9) 5.0% - ---------------------------------------------------------------------------------- 12. TYPE OF REPORTING PERSON (See Instructions) IA - ---------------------------------------------------------------------------------- Page 2 of 4 Pages

SCHEDULE 13G - ------------ Item 1(a) Name of Issuer: Halliburton Company 1(b) Address of Issuer's Principal Executive Offices: 3600 Lincoln Plaza 500 N. Akard Street Dallas, TX 75201 Item 2(a) Name of Person Filing: Barrow, Hanley, Mewhinney & Strauss, Inc. 2(b) Address of Principal Business Office or, if none, Residence: One McKinney Plaza 3232 McKinney Avenue, 15th Floor Dallas, TX 75204-2429 2(c) Citizenship: A Nevada corporation 2(d) Title of Class of Securities: Common Stock 2(e) CUSIP Number: 406216101 Item 3 If this statement is filed pursuant to Rules 13d-1(b), or 13d-2(b): The reporting person is an Investment Adviser registered under Section 203 of the Investment Advisers Act of 1940. Item 4 Ownership: 4(a) Amount beneficially owned: 22,045,960 shares 4(b) Percent of Class: 5.0% 4(c) Number of shares as to which such person has: (i) Sole power to vote or to direct the vote: 4,169,700 shares Page 3 of 4 Pages

(ii) Shared power to vote or to direct the vote: 17,876,260 shares (iii) Sole power to dispose or to direct the disposition of: 22,045,960 shares (iv) Shared power to dispose or to direct the disposition of: -- Item 5 Ownership of Five Percent or Less of a Class: Not Applicable. Item 6 Ownership of More than Five Percent on Behalf of Another Person: The right to receive or the power to direct the receipt of dividends from, or the proceeds from the sale of, the common stock is held by certain clients of the reporting person, none of which has such right or power with respect to five percent or more of the common stock. Item 7 Identification and Classification of the Subsidiary which Acquired the Security Being Reported on by the Parent Holding 3600 Lincoln Plaza: Not Applicable. Item 8 Identification and Classification of Members of the Group: Not Applicable. Item 9 Notice of Dissolution of Group: Not Applicable. Item 10 Certification: By signing below the undersigned certifies that, to the best of its 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 purpose or effect. After reasonable inquiry and to the best of its knowledge and belief, the undersigned certifies that the information set forth in this statement is true, complete and correct. BARROW, HANLEY, MEWHINNEY & STRAUSS, INC. By: /s/ Bryant M. Hanley, Jr. Name: Bryant M. Hanley, Jr. Title: President February 11, 1999 Page 4 of 4 Pages