EXHIBIT (32a)

                CERTIFICATION 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 Unifi, Inc. (the "Company") Annual Report on Form 10-K
for the period ended June 27, 2004 as filed with the Securities and Exchange
Commission on the date hereof (the "Report"), I, Brian R. Parke, Chairman of the
Board, President and 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, as
                  amended; and

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



Date:  September 17, 2004       By:  /s/ BRIAN R. PARKE
       ------------------            ---------------------------------------
                                     Brian R. Parke
                                     Chairman of the Board,
                                     President and
                                     Chief Executive Officer