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 1, 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 1, 1998 registrant issued a press release entitled  Halliburton
 Renotifies   European  Commission   pertaining,   among  other  things,  to  an
 announcement  that  registrant has  renotified  the European  Commission of its
 proposed merger with Dresser Industries. Registrant's original notification was
 incomplete  in  certain  respects.  Registrant  has  now  supplied  all  of the
 information thus far requested by the European Commission.  As a consequence of
 the renotification,  the initial one month review period provided for under the
 European Commission's merger regulations will now expire on July 2, 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 1, 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 1, 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 1, 1998
                           Incorporated by Reference





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


FOR IMMEDIATE RELEASE               Contact:Guy T. Marcus
June 1, 1998                                Vice President-Investor Relations
                                            (214) 978-2691


                   HALLIBURTON RENOTIFIES EUROPEAN COMMISSION


         DALLAS, Texas -- Halliburton Company (NYSE:HAL) today announced that it
has renotified its proposed merger with Dresser  Industries,  Inc.  (NYSE:DI) to
the European Commission.  Halliburton's  original notification was incomplete in
certain  respects.  Halliburton has now supplied all of the information thus far
requested by the European  Commission.  As a consequence of the  renotification,
the initial one month review period provided for under the European Commission's
merger regulations will now expire on July 2, 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