PLT LINED UP ON WRONG ILS.

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ON INITIAL CONTACT WITH APCH CTL; WE WERE CLRED FOR THE ILS RWY 36R AT MEM. I REQUESTED A CAT III APCH FOR AN EQUIP CHK AND I WAS TOLD TO CONTACT THE FINAL APCH CTLR FOR MY REQUEST. I CONTACTED THE FINAL CTLR; REQUESTED A CAT III APCH; AND I WAS GIVEN A CAT III APCH. WE WERE CLRED TO INTERCEPT THE LOC AND I ASSUMED THAT WE WERE CLRED FOR RWY 36L CAT III. AT APPROX 2100 FT MSL; THE CTLR GAVE US AN IMMEDIATE R TURN TO INTERCEPT RWY 36R ILS. WE IMMEDIATELY TURNED TO INTERCEPT RWY 36R AND WE WERE CLRED TO LAND. PREVIOUSLY; RWY 36R WAS NOTAM NOT AUTH FOR CAT III AND CURRENT NOTAMS HAD THE GS OTS DAILY UNTIL XA30 LCL. I GENERALLY INCLUDE THE RWY TO WHICH I WAS CLRED FOR AS A PART OF MY READBACK TO APCH. IF I HAD ACCOMPLISHED THIS PROC; I COULD HAVE PREVENTED THIS MISUNDERSTANDING. SUPPLEMENTAL INFO FROM ACN 291879: THE ONLY CAT III RWY AVAILABLE WAS RWY 36L. I CANNOT POSITIVELY SAY THAT WE WERE CLRED TO INTERCEPT THE RWY 36L LOC BUT WE WERE CLRED FOR THE CAT III APCH. UPON INTERCEPTING THE LOC FOR RWY 36L WE WERE TOLD THAT OUR RWY WAS RWY 36R.