CL65-700 CREW CLBED ACFT TOO HIGH FOR THE ACFT GROSS WT; GOT ON THE BACK SIDE OF THE PWR CURVE; AND WITH SLOWING AIRSPD; HAD TO DSND TO PREVENT A STALL IN ZKC CLASS A.

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FLT WAS PROCEEDING FROM CVG; CRUISING AT FL350 AT .77 MACH. AT THE TIME OF THE EVENT; THE FLT'S POS WAS APPROX 85 MI E OF SGF. DUE TO A LINE OF WX EXTENDING FROM OMA TO OKC; THE FLT REQUESTED AND WAS GRANTED FL390. PRIOR TO THE ALT REQUEST; THE CREW REFERRED TO AND CONSULTED THE 'MAX WT TO ALT CHART' WHICH STATED THE ACFT; UNDER CURRENT WT AND ISA CONDITIONS; COULD SAFELY ACHIEVE FL390. THE FLT CLBED TO FL390 WITHOUT EVENT. AFTER ARRIVING AT FL390; THE CREW REALIZED THE ACFT WOULD NOT ACCELERATE; AND SLOWLY BEGAN TO DECELERATE. THE CREW AT ONCE NOTIFIED ZKC; REQUESTED A LOWER ALT. ZKC'S RESPONSE WAS THAT THEY WERE UNABLE LOWER DUE TO ONCOMING TFC. AS THE ACFT CONTINUED TO SLOW OVER NEXT FEW MINS; THE CREW CONTINUED TO MAKE REQUESTS FOR A LOWER ALT. DURING 1 REQUEST; THE CTLR SAID WE WOULD HAVE TO DECLARE AN EMER TO GET LOWER. A TOTAL OF 5 REQUESTS WERE MADE FOR A LOWER ALT STATING THAT WE WOULD NOT BE ABLE TO MAINTAIN CURRENT ALT BECAUSE OF ACFT AIRSPD DEGRADATION. THE CREW ALSO AT THIS TIME HAD THE TCASII SET ON A 20 MI RANGE AND DID NOT DISPLAY ANY CONFLICTING TFC. PRIOR TO A STALL WARNING; WE HAD NO OTHER CHOICE THAN TO START A SLOW TURNING DSCNT TO THE R. FINALLY; CTR CAME BACK WITH AN ALT ASSIGNMENT OF FL370 FT AND A TURN TO 360 DEGS. SHORTLY THEREAFTER; WE RECEIVED ANOTHER HEADING CHANGE TO 270 DEGS AND TOLD TO CHANGE TO ANOTHER FREQ. WITHOUT FURTHER DISCUSSION; THE FLT CONTINUED ON TO DEST WITHOUT EVENT. THIS EVENT WOULD NOT HAVE HAPPENED IF WX HAD NOT BEEN IN THE AREA; ONE LESS AIRPLANE HAD NOT BEEN IN THE VICINITY OR IF THE CTLR HAD ONE LESS ACFT TO MANAGE. ON ANOTHER DAY; AT THE SAME TIME; AT THE SAME PLACE; THE CTLR WOULD HAVE BEEN GLAD TO GIVE THE CREW ANOTHER ALT.