Heya, just wanted to chime in I *just* did this with my G1WH using
this video of a guy doing it for a G1W as the general guide/template, and just wanted to put this here to help out any other wayward souls in the same situation that I was in. The battery in my cam swelled up, and wouldn't hold a charge anymore.. fortunately it'd still record, but the date was reset to Jan 1st, 2013 on a regular basis and every so often it'd corrupt one of the looped recorded files and reboot itself. Annoying to say the least.
In the video the layout of my G1WH was essentially the same, but be wary of the front sensor wiring as it is a little fragile (one of the leads came off while I was cracking the case... I might have been a little overzealous with my pry tool, so I had to re-solder that too).
In addition, in the description he links a 400 mAh lipo battery that fits/works just fine as a replacement, available from a seller on ebay. The bigger battery fits in the case perfectly, including a thicker new piece of doublesided tape underneath it to hold it in place. Worked like a charm for me, pretty much exactly as demo'd in the youtube video I linked.
Just in case you want the original, smaller battery's info, on mine under the doublesided tape it was attached to I've got: 402030, 3.7V, 180 mAh.
It has been my experience that searching ebay for "402030 battery" will give you plenty of options if you wanted to stick with the original... but, I recommend the 400 mAh battery upgrade, because why not go for something more robust while you've got the guts exposed anyway. Edit: now with a picture I forgot I snapped for posterity's sake.
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Hope this helps!