AN A320 APCHING MDW RESPONDED TO A VFR RA CONFLICTING WITH AN ATC COMMAND.

Narrative

AT 4000 FT ON AN EASTERLY HDG TO INTERCEPT THE RWY 4R LOC AT MDW; MDW APCH CALLED VFR TFC AT 1 O'CLOCK POS; 5 MI; ALT 4000 FT; TYPE UNKNOWN. THE CTLR SAID 'LET ME KNOW WHEN YOU HAVE HIM IN SIGHT.' THE BASES OF THE CLOUDS WERE APPROX 3600 FT AND VISIBILITY WAS GOOD LOOKING DOWN AT AN ANGLE BUT WAS 0 FT HORIZLY. WE LOOKED FOR THE ACFT BUT COULD NOT SEE IT BECAUSE OF THE CLOUDS. WE DID SEE IT ON TCASII AND IT HAD ALREADY TURNED INTO A WHITE TARGET. I TOLD THE CTLR WE WERE IN THE CLOUDS AND COULD NOT SEE THE TFC; BUT HE WAS VERY BUSY AND I AM NOT SURE IF HE HEARD ME. I TOLD MY FO TO GET READY FOR AN 'RA' AND TOLD THE CTLR WE WERE IN THE CLOUDS AND COULD NOT SEE THE TFC. THE CTLR GAVE US A 10 DEG TURN TO THE L WHICH WE COMPLIED WITH. AT THIS POINT; THE TFC WAS ONLY 2 MI AWAY AND CLOSING. THE CTLR ISSUED AN IMMEDIATE L 140 DEG TURN AND A CLB TO 5000 FT. AT THIS VERY MOMENT; THE TCASII GAVE AN RA TO 'DSND.' I TOLD THE CTLR WE WERE COMPLYING WITH AN RA. MY FO HAD ALREADY DISCONNECTED THE AUTOPLT AND AUTOTHROTTLES AND WAS DSNDING AND TURNING L. THE TFC PASSED OVERHEAD ON TCASII AT +100 FT. WE CONTINUED OUR DSCNT UNTIL TCASII SAID CLR OF CONFLICT AND THEN WE CLBED BACK TO 4000 FT. I THOUGHT WE WERE GOING TO HIT; ALTHOUGH I NEVER SAW THE TFC. I TOLD THE CTLR THAT IT WAS VERY CLOSE! HE CLRED US FOR THE VISUAL TO RWY 4R AND WE LANDED WITHOUT FURTHER INCIDENT. SUPPLEMENTAL INFO FROM ACN 668964: JUDGING OFF TCASII WE CLRED BY 150-200 FT. 500-600 FT ALT WAS LOST AND WE CLBED BACK TO 4000 FT TO INTERCEPT THE ARR.