Charge indicator on, battery doesn't charge

My iPad 3 wi-fi has a problem charging. I would plug it in using the original cable and power adapter and it woudln't take a charge. I brought it to the Apple store and they tried several cables and got the same result. They told me it was dead and I needed to buy a new one.

I bought a new dock connector and installed it yesterday. The "lightning bolt"charge indicator comes up but after having it plugged in for 3 hours the battery didn't charge at all.

Is this a problem with the battery? Any advice is appreciated. I hate to buy a battery if I don't need one.