Exhibit 32.1

                    CERTIFICATION OF CHIEF EXECUTIVE OFFICER
                       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  Quarterly  Report of Brigham Exploration Company
(the"Company")  on  Form 10-Q for the period ending March 31, 2005 as filed with
the Securities  and Exchange Commission on the date hereof (the "Report"), I,
Ben M.Brigham,  President,  Chief  Executive  Officer and Chairman of the Board
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 (15 U.S.C. 78m or
          78o(d)); and

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

Dated:  May 6, 2005                /s/ BEN M. Brigham
                                   ---------------------------------------------
                                   Ben M. Brigham
                                   Chief Executive Officer, President and
                                   Chairman of the Board


     This certification shall not be deemed to be "filed" for the purpose of
Section 18 of the Securities Exchange Act of 1934, as amended, and will not be
incorporated by reference into any registration statement filed under the
Securities Act of 1933, as amended, unless specifically identified therein as
being incorporated therein by reference.

     A signed original of this written statement required by Section 906, or
other document authenticating, acknowledging, or otherwise adopting the
signature that appears in typed form within the electronic version of this
written statement required by Section 906, has been provided to Brigham
Exploration Company and will be retained by Brigham Exploration Company and
furnished to the Securities and Exchange Commission or its staff upon request.