Cybershot will not store clock settings

My Sony Cybershot (DSC-W150) has suddenly stopped keeping the correct date and time after about seven months of working without a problem. I've tried various suggestions - formatting the internal memory, using the "initialize" option to restore the default settings, using a new battery - all of which have done nothing. Most of the time it just defaults to January 1, 2008 and then asks me to set the clock every time I turn it on. It still takes pictures fine, but will not organize them by date. Plus, it's a nuisance to set the clock every time I turn it on.

I have had success a couple times getting the correct date to save, but the clock does not appear to run after that. Example: if I set the clock at 1:30 and check it again at 1:40, the camera still reads 1:30. I assume if I left it off for a day, the next day it would tell me that it's yesterday. When I try to reset the time, it loses the settings again (1/1/08, 12midnight).

Sony Customer Support was basically useless, as they only suggested things I TOLD them I tried and also to plug in the AC adapter which the Cybershot doesn't even have! They said if their "fixes" don't work I will need to send the camera out for service.

One other detail: even after replacing the battery with a brand new one, the battery indicator on the LCD shows it as partially drained (battery w/ hash marks through it). After charging, the first battery never showed this immediate drainage until this clock setting fiasco started. This is why I thought replacing the battery would help, which it didn't, but these problems are likely related.

Question: Does the Cybershot contain a separate, internal lithium battery (like a small watch battery?) for the clock? Sony seems to want to keep it a secret, and I haven't opened the camera to find out. I've heard of some cameras that have this, and if it's something I could replace myself without sending my camera out I would appreciate that knowledge!