SECURITIES AND EXCHANGE COMMISSION

                             Washington, D.C. 20549



                                    FORM 8-K

                                 CURRENT REPORT

                     PURSUANT TO SECTION 13 OR 15(d) OF THE
                         SECURITIES EXCHANGE ACT OF 1934

                DATE OF REPORT (date of earliest event reported)

                                  JUNE 12, 1998

                               Halliburton Company
             (Exact name of registrant as specified in its charter)

State or other                 Commission                IRS Employer
jurisdiction                   File Number               Identification
of incorporation                                         Number

Delaware                         1-3492                  No. 75-2677995

                               3600 Lincoln Plaza
                             500 North Akard Street
                            Dallas, Texas 75201-3391
                    (Address of principal executive offices)

                         Registrant's telephone number,
                       including area code - 214/978-2600






                                Page 1 of 5 Pages
                       The Exhibit Index Appears on Page 4


         INFORMATION TO BE INCLUDED IN REPORT

Item 5.  Other Events

         The registrant  may, at its option,  report under this item any events,
with respect to which information is not otherwise called for by this form, that
the registrant deems of importance to security holders.

         On June 12, 1998 registrant issued a press release entitled Halliburton
 Renotification   Effective  Date   pertaining,   among  other  things,   to  an
 announcement that following further contacts with the European Commission,  the
 renotification became complete on June 3, 1998 with the result that the initial
 one-month  review  period  provided for under the European  Community's  merger
 regulation will in principle expire on July 6, 1998.

Item 7.  Financial Statements and Exhibits

         List below the financial  statements,  pro forma financial  information
and exhibits, if any, filed as part of this report.

         (c)      Exhibits.

                  Exhibit 20 - Press release dated June 12, 1998.







                                Page 2 of 5 Pages
                       The Exhibit Index Appears on Page 4



                                   SIGNATURES

         Pursuant to the  requirements  of the Securities  Exchange Act of 1934,
the  registrant  has duly  caused  this report to be signed on its behalf by the
undersigned hereunto duly authorized.


                                      HALLIBURTON COMPANY




Date:    June 23, 1998                By:
                                         ----------------------------------
                                         Susan S. Keith
                                         Vice President and Secretary







                                Page 3 of 5 Pages
                       The Exhibit Index Appears on Page 4



                                  EXHIBIT INDEX



Exhibit                                                       Sequentially
Number                     Description                        Numbered Page

20                         Press Release of                   5 of 5
                           June 12, 1998
                           Incorporated by Reference





                                Page 4 of 5 Pages
                       The Exhibit Index Appears on Page 4



FOR IMMEDIATE RELEASE               Contact: Guy T. Marcus
June 12, 1998                                V.P.-Investor Relations
                                             214-978-2691


                    HALLIBURTON RENOTIFICATION EFFECTIVE DATE


         DALLAS,  Texas  -  On  June  1,  1998  Halliburton  Company  (NYSE:HAL)
announced that it had renotified  its proposed  merger with Dresser  Industries,
Inc. (NYSE: DI) to the European Commission.  Following further contacts with the
Commission,  the renotification  became complete on June 3, 1998 with the result
that the  initial  one-month  review  period  provided  for under  the  European
Community's merger regulation will in principle expire on July 6, 1998.
         Halliburton  Company is one of the world's largest  diversified  energy
services, engineering, maintenance, and construction companies. Founded in 1919,
Halliburton  provides a broad range of energy services and products,  industrial
and marine engineering and construction services.


                                       ###


                                Page 5 of 5 Pages
                       The Exhibit Index Appears on Page 4