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                           May 18, 2021

       Lawrence Bloch
       President
       Infinity Pharmaceuticals, Inc.
       1100 Massachusetts Avenue, Floor 4
       Cambridge, Massachusetts 02138

                                                        Re: Infinity
Pharmaceuticals, Inc.
                                                            Registration
Statement on Form S-3
                                                            Filed May 13, 2021
                                                            File No. 333-256096

       Dear Dr. Bloch:

              This is to advise you that we have not reviewed and will not
review your registration
       statement.

               Please refer to Rules 460 and 461 regarding requests for
acceleration. We remind you
       that the company and its management are responsible for the accuracy and
adequacy of their
       disclosures, notwithstanding any review, comments, action or absence of
action by the staff.

                                                        Please contact Jane
Park at 202-551-7439 with any questions.




                           Sincerely,


                           Division of Corporation Finance

                           Office of Life Sciences
       cc:                                              Cynthia Mazareas, Esq.