EXHIBIT 31.1

                    CERTIFICATION OF CHIEF EXECUTIVE OFFICER
                        PURSUANT TO RULE 13A-14(A) OF THE
                         SECURITIES EXCHANGE ACT OF 1934

I, Bud M. Brigham, certify that:

1.   I have reviewed this quarterly report on Form 10-Q of Brigham Exploration
     Company;

2.   Based on my knowledge, this report does not contain any untrue statement of
     a material fact or omit to state a material fact necessary to make the
     statements made, in light of the circumstances under which such statements
     were made, not misleading with respect to the period covered by this
     report;

3.   Based on my knowledge, the financial statements, and other financial
     information included in this report, fairly present in all material
     respects the financial condition, results of operations and cash flows of
     the registrant as of, and for, the periods presented in this report;

4.   The registrant's other certifying officer(s) and I are responsible for
     establishing and maintaining disclosure controls and procedures (as defined
     in Exchange Act Rules 13a-15e and 15d-15(e)) for the registrant and have:

     a)   Designed such disclosure controls and procedures or caused such
          disclosure controls and procedures to be designed under our
          supervision, to ensure that material information relating to the
          registrant, including its consolidated subsidiaries, is made known to
          us by others within those entities, particularly during the period in
          which this report is being prepared;

     b)   Evaluated the effectiveness of the registrant's disclosure controls
          and procedures and presented in this report our conclusions and about
          the effectiveness of the disclosure controls and procedures, as of the
          end of the period covered by this report based on such evaluation; and

     c)   Disclosed in this report any change in the registrant's internal
          control over financial reporting that occurred during the registrant's
          most recent fiscal quarter (the registrant's fourth fiscal quarter in
          the case of an annual report) that has materially affected, or is
          reasonably likely to materially affect, the registrant's internal
          control over financial reporting; and

5.   The registrant's other certifying officer(s) and I have disclosed, based on
     our most recent evaluation of internal control over financial reporting, to
     the registrant's auditors and the audit committee of the registrant's board
     of directors (or persons performing the equivalent functions):

     a)   All significant deficiencies and material weaknesses in the design or
          operation of internal control over financial reporting, which are
          reasonably likely to adversely affect the registrant's ability to
          record, process, summarize and report financial information; and

     b)   Any fraud, whether or not material, that involves management or other
          employees who have a significant role in the registrant's internal
          control over financial reporting.



Date:  November 9, 2004

          /s/ BUD M. BRIGHAM
          -----------------------------------------
          Bud M. Brigham
          Chief Executive Officer, President and
          Chairman of the Board


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