CERTIFICATION PURSUANT TO SECTION 906 OF THE SARBANES-OXLEY ACT

I, Carter E.  Anthony,  President of RMK High Income Fund,  Inc.  (the  "Fund"),
certify that:

      1. The Form  N-CSR of the Fund  (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 Fund.



Date: June 2, 2004                              /s/ Carter E. Anthony
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                                                Carter E. Anthony, President



         CERTIFICATION PURSUANT TO SECTION 906 OF THE SARBANES-OXLEY ACT

I, Joseph C.  Weller,  Treasurer  of RMK High Income Fund,  Inc.  (the  "Fund"),
certify that:

      1. The Form  N-CSR of the Fund  (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 Fund.



Date: June 2, 2004                             /s/ Joseph C. Weller
     ------------------------           ----------------------------------------
                                               Joseph C. Weller, Treasurer