Samsung sgh-a707 won't stay on. Shuts off when batt seems full.

Hello,
So I went to bed last night and my phone had some battery in it. I woke up this morning and while reading a text from my friend my phone just turned off. It was also really hot. So then I plugged it into the charger and tried to turn it back on and it wouldn't go so I unplugged it, took out the battery and held the power button down to try to drain all excess battery or whatever that process is called. Then, an hour later, I plugged in the charger and put the battery in and it turned on and it worked. I then unplugged the charger and the battery was full and then just a short time later, it shut off again. I plugged it into the charger and tried to turn it on again, but it wouldn't. So then I started typing this problem and mid-way through it beeped and there's this big orange battery on my screen and all three bars are filled in. Also, there were no battery low notifications. Hey, it just beeped again, showed an empty battery and then went back to the screen with the full battery.