I had the same problem myself.
The fix was to replace the battery in the phone. I ordered a replacement from Amazon.com. As I also had had to replace the screen just before that (my wife face planted the phone on a rock), I already possessed the required tools to take it apart - they came with the replacement screen.
It needs a very small hex driver, a very small philips driver, a spudger and the SIM pin (or something similarly pointy, such as a needle, to remove the really long locking ping that runs the length of one side of the phone)