Not knowing the history of how deeply you discharge the battery, or how often you charge it, it's hard to say what the problem is. For starters, read the other recent thread about the PRS350 battery.
It is possible that your usage pattern has led to such a short lifespan on the battery. It is also possible that you are just unlucky and got a bum battery. Heat will also greatly shorten the life of a Lithium battery. They love the cold, however.
If you feel you have nothing to lose, you might try JUST ONE deep discharge and full recharge cycle to see if things improve. If you do this, make sure to charge the battery longer than you think necessary (overnight is good). If this procedure works, don't repeat it, but do follow the recommendations in the other thread.
If you need a new battery, there are places to buy one online. I'm sure someone else on MR can tell you how to open your reader. OTOH, now you have an excuse to buy a new PRS650
Joe