iPhone turns off when not charging

My iPhone, which I just bought yesterday, worked fine when I first got it.
So I thought it was fully charged, but then it just suddenly cut off on me, so I began to charge it through the wall outlet.
Then when the battery said fully charged I unplugged it, and about a minute later it just cut off again.
After that it wouldn't even show the apple logo when turning on unless I had it plugged in.
Then it began not even turning on when plugged in.
So now I have it sitting in the dock, it is off and won't come on, and I'm waiting for the restore software to download, because I did happen to get it into restore mode for I guess a second, but long enough for it to recognize the phone in restore mode I suppose.
So is this going to work, or am I going to have to go get a new one?