<PAGE>

                                                                    EXHIBIT 31.1

                                  CERTIFICATION

I, Zafiris G. Zafirelis, certify that:

1.    I have reviewed this quarterly report on Form 10-Q of Biopure Corporation;

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- 15(e) 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 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: June 9, 2005

/s/ Zafiris G. Zafirelis
---------------------------
Zafiris G. Zafirelis
President and Chief Executive Officer