Without having the phone here I couldn't say for sure, but it sounds as if something is constantly draining the battery. While the battery is charging it doesn't matter - it just charges a bit slower. Once it's fully charged, the charging circuit shuts off and he battery starts draining.
Is it some software using tons of current? Is it a hardware short? Is it a partial short inside the battery? That's impossible to tell from here. One quick check you can make is find someone with an S3 and use their battery. If the problem stops, you need a new battery. Otherwise it's a software or hardware problem. (And if I were a betting man, I'd bet on hardware, which means a trip to a repair facility.)