Mete is right and wrong. Both cartridges use the same .243" bullets, and the head diameter is the same so the bolt would not be changed. 6mm is a longer cartridge OAL and might or might not fit into and feed from the .243 magazine.
BUT a 6mm Remington reamer will not "clean up" a .243 Winchester chamber.
If the rifle of interest has a heavy barrel, it could be set back, rethreaded and rechambered. Otherwise the only route would be a new barrel, either on the .243 or on a "donor" action like Countryboy's. Either would be expensive and gain you only a little velocity. And you would have a rifle for an obsolescent cartridge with lower availability of ammunition and a lowered resale value.
Bad idea.
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