EX-99.906 CERT


         CERTIFICATION PURSUANT TO SECTION 906 OF THE SARBANES-OXLEY ACT

I, Carter E.  Anthony,  President and  Principal  Executive  Officer 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:  May  27, 2005                  /s/ Carter E. Anthony
     ---------------                  ---------------------
                                      Carter E. Anthony, President and Principal
                                      Executive Officer



         CERTIFICATION PURSUANT TO SECTION 906 OF THE SARBANES-OXLEY ACT

I, Joseph C.  Weller,  Treasurer  and  Principal  Financial  Officer 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:  May  27, 2005                  /s/ Joseph C. Weller
     ---------------                  --------------------
                                      Joseph C. Weller, Treasurer and Principal
                                      Financial Officer