EXHIBIT (99b)


                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 year ended June 30, 2002 as filed with the Securities and
Exchange Commission on the date hereof (the "Report"), I, Willis C. Moore, III,
Chief Financial 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 23, 2002                By: WILLIS C. MOORE, III
       ---------------------------           -----------------------------------
                                             Willis C. Moore, III
                                             Executive Vice President and
                                             Chief Financial Officer