A DC9-10 CREW DSNDS OUT OF FL240 UPON 'RECEIVING A CLRNC TO FL200.' THE CTLR CAUGHT THEM AT FL230 AND ASSIGNED THAT AS A 'STOP' ALT; DISPUTING THEIR CLRNC TO FL200.

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FLYING RIVERS ARR STL LAMBERT ARPT AT FL240; LIGHT TURB BEGAN TO MAKE FLT UNCOMFORTABLE; SO WE ASKED ZKC IF LOWER ALT WAS SMOOTHER. CTLR WAS VERY BUSY AND DID NOT RESPOND IMMEDIATELY; SO I PICKED UP THE PA AND WAS MAKING AN ANNOUNCEMENT ABOUT THE CHOPPINESS. IN THE MIDDLE OF MY ANNOUNCEMENT ATC TOLD US WE HAD TO GO TO FL200 OR BELOW FOR SMOOTHER AIR AND THE FO AND I BOTH HEARD HIM CLR US TO FL200. I BEGAN A DSCNT AND FINISHED THE PA ANNOUNCEMENT; AS THE FO REPEATED THE CLRNC OVER THE RADIO TO ATC. THE CTLR MADE NO CORRECTION AND IMMEDIATELY BEGAN TALKING TO ANOTHER ACFT. AS I WAS PASSING FL230 THE CTLR SAID HE NEEDED US AT FL240 AND THAT HE HAD ONLY TOLD US THE ALT WE NEED TO DSND TO FOR SMOOTH AIR; NOT AN ACTUAL CLRNC. THE FO RESPONDED BY TELLING HIM WE HAD BEEN CLRED TO FL200. THE CTLR TOLD US TO MAINTAIN FL230 AND TURNED US OVER TO ANOTHER ZKC FREQ WITH NO FURTHER DISCUSSION. NO CONFLICT WAS OBSERVED ON OUR PART AND WE COMPLETED THE FLT WITH NO FURTHER OCCURRENCE. CONTRIBUTING FACTORS -- IF WE IN FACT HAD MADE AN ERROR (WHICH I DON'T THINK WE DID): 1) CTLR WAS SO BUSY HE MISSED THE FO'S RESPONSE TO HIS UNDERSTOOD CLRNC TO DSND TO A LOWER ALT. 2) MY CONCENTRATION WAS DIVIDED BTWN COMS WITH ATC AND MAKING THE PA ANNOUNCEMENT.