Encore Acquisition 10-Q 2004 Q2 Quarterly report

Exhibit 32.1

Section 1350 Certification

     In connection with the Quarterly Report of Encore Acquisition Company (the “Company”) on Form 10-Q for the period ending June 30, 2004 as filed with the Securities and Exchange Commission on the date hereof (the “Report”), I, I. Jon Brumley, 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 Exchange Act of 1934; and

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

     Date: August 6, 2004

/s/ I. Jon Brumley

I. Jon Brumley
Chairman and Chief Executive Officer

     A signed original of this written statement required by Section 906 has been provided to the Company and will be retained by the Company and furnished to the Securities and Exchange Commission or its staff upon request.

     Note: The certification of the registrant furnished in this exhibit is not deemed “filed” for purposes of Section 18 of the Securities and Exchange Act of 1934, as amended, or otherwise subject to the liabilities of that Section. Registration Statements or other documents filed with the Securities and Exchange Commission shall not incorporate this exhibit by reference, except as otherwise expressly stated in such filing.