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

This certification is provided pursuant to Sec. 906 of the Sarbanes-Oxley Act of 2002, 18 U.S.C. Sec. 1350, and accompanies the Annual Report on Form 10-K/A Amendment No. 2 for the period ended December 31, 2003 of Halliburton Company (the "Company") 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 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                    

David J. Lesar

Chief Executive Officer

Date: May 11, 2004________