Same problem as many others - phone shuts down randomly at 70% power. Lots of blather on the web about why (new iOS, etc. etc.) but the only solution that appeared to actually solve anyone's problem was battery replacement.
No problems with the replacement. Pay attention to where the cable under connector 3 actually is before you release it. When I released the little white flap the whole top came off and I didn't really see where the cable had come from. After some fussing on the rebuild side though I got it into the right place. Hardest part is getting the logic board with the camera attached all back into the correct place. That took a while; just stick with it and use that spudger to push the camera down into place when you get it oriented correctly. Although I bought a 4S before attempting this repair not being too confident in my computer repair abilities, I now have an operational 3GS that I'm hoping to get ATT to unlock so I can use it overseas with prepaid SIM cards from local carriers. That part may be harder than the battery replacement!
Be brave. Keep the screws separate in the same pattern as where you took them out of the phone.