When the engine is running, the voltage at the battery should be up above 13 volts, above 1200 RPM, it should be near or just above 14 volts. If it is not, then the alternator is not charging. You stated the car ran, but then died, and that (the way I read your post) the battery voltage fell with the engine running, enough that the engine quit.

Sounds like a bad alternator to me - and after the abuse this has put the battery thru, I'd guess you are looking at both a new alternator AND battery - but this is all just based on a few words you typed.