

                                                                EXHIBIT 5.1


                     [LETTERHEAD OF VINSON & ELKINS L.L.P.]




   (713) 758-2222                                               (713) 758-2346



                                       June 1, 1998

Halliburton Company
3600 Lincoln Plaza
500 North Akard Street
Dallas, Texas 75201-3391

Ladies and Gentlemen:

         We acted as counsel to Halliburton Company, a Delaware corporation (the
"Company"),  in connection  with the  preparation of the Company's  Registration
Statement on Form S-8 (the "Registration  Statement") to be filed by the Company
with the  Securities and Exchange  Commission  pursuant to the Securities Act of
1933, as amended, which Registration Statement relates to the offering, sale and
delivery  of  (i) an aggregate of up to 1,000,000 shares of the Company's common
stock,  par  value  $2.50 per  share (the "Shares"), pursuant to the Halliburton
Profit Sharing and Savings Plan and the Brown & Root, Inc. Employees' Retirement
and Savings  Plan (the "Plans"),  and (ii) the interests  of participants in the
Plans (the "Interests").

         Before  rendering this  opinion, we have such certificates, instruments
and documents and reviewed such  questions of law as we considered  necessary or
appropriate  for the purposes of  this opinion.  In  addition,  we relied  as to
factual  matters  on  certificates  or  other communications  of officers of the
Company.

         Based upon the foregoing  examination and review, we are of the opinion
that the Shares and the Interests  have been duly  authorized  for issuance and,
when the Registration  Statement has been declared  effective and the Shares and
the  Interests are issued in accordance  with the  provisions of the Plans,  the
Shares will be validly issued,  fully paid  and  nonassessable and the Interests
will be validly issued.

         This opinion is rendered as of the effective  date of the  Registration
Statement.  We hereby consent to the filing of this opinion as an exhibit to the
Registration Statement. In giving this consent,  however, we do not hereby admit
that we are within the  category  of persons  whose  consent is  required  under
Section  7 of the  Securities  Act of  1933,  as  amended,  and  the  rules  and
regulations of the Securities and Exchange Commission thereunder.

                                       Very truly yours,




                                       VINSON & ELKINS L.L.P.



