Battery seems not to hold charge.

I've got a black 32GB iPhone 3GS that I've bought as faulty on eBay. I've managed to kick it off of recovery loop and properly restore it, and now it turns on nicely and every feature works well.

The problem is, that it doesn't charge. I can not start the phone without plugging it in to power, and when I do, it boots, and I can use it for approximately 3 minutes - with power attached - after which it would restart. If I remove the cable after booting it, it would shut down very soon. Also, if I leave it plugged in, it would heat up a little bit, unlike my other, properly working 3GS. The battery percentage shows 0 or 1%.

Interestingly, iTunes shows that it's charging for a minute or so, and afterwards it only shows an empty battery icon near the phone's name.

Do you think it's the battery, or can it be some other stupid software issue?

Thanks for you time!

Yeah, replacing the battery did solve my problem.