How do I repair broken bathtub faucet handles?
Its likely the whole cartridge will need replacement. With it you will get a new handle stem.  

This might be a sign to replace the whole unit.  If you have a three handle setup, a cover plate will let you upgrade to any single handle style.  If you go this route, be sure to get a tub filler spout WITH diverter (the single handle tub/shower valves usually don't have diverters.