I found the Sony PRS-650 Service Manual somewhere on the Internet and saved it. It shows how to change the battery - did not look too difficult. Problem is what battery to use? The manual quotes a Sony part number for the battery and I went to Sony's parts site and checked that out - not in stock but like US $55 for special order.
I figure when enough PRS-650's start experiencing battery failure some enterprising battery company will offer a replacement for a much more reasonable price and probably document the replacement procedure as well, as several companies have for the PRS-505.
BTW there is an elegant video on YouTube for replacing the battery in the PRS-505.
My PRS-650 has been working perfectly - but I do worry about battery failure, since replacement by Sony is around $100.