What was found
AT FL 410 RECEIVED THE EMERGENCY PRESS LIGHT. RAN THE CHECKLIST AND RESET THE SYSTEM BACK TO NORMAL. THE CABIN ALTITUDE WENT TO 9000 FEET. THE RATE CONTROLLER SEEMED TO HAVE NO AFFECT AND COULD NOT GET THE CABIN ALTITUDE TO COME DOWN TO THE PROPER ALTITUDE. DESCENDED TO FL 390 AND THE CABIN STABILIZED AT 0200 FEET AND 8.0 DIFF. UPON DIVERSION TO KICT THE CABIN RATE SUDDENLY JUMPED TO 6000 FPM AND THE CABIN ALT IMMEDIATELY WENT THRU 10,000 FEET. DECLARED AN EMERGENCY AND DIVERTED INTO KLIT. CAUSE TRACED TO FAILED ACM OVERTEMP SWITCH.