Exhibit 32.1

                            CERTIFICATION PURSUANT TO
                             18 U.S.C. SECTION 1350
                             AS ADOPTED PURSUANT TO
                  SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

In connection with the amended Quarterly Report of Halliburton Company (the
"Company") on Form 10-Q/A for the period ending March 31, 2003 as filed with the
Securities and Exchange Commission on the date hereof (the "Report"), I, David
J. Lesar, Chief Executive Officer of the Company, certify, pursuant to 18 U.S.C.
Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of
2002, that:

         (1)      The Report fully complies with the requirements of section
                  13(a) or 15(d) of the Securities Act of 1934; and

         (2)      The information contained in the Report fairly presents, in
                  all material respects, the financial condition and results of
                  operations of the Company.

         /s/ David J. Lesar
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         David J. Lesar
         Chief Executive Officer
         January 15, 2004