I have the G11. There is a "Troubleshooting" section near the back of the owner's manual. For your symptom, it only says to be sure the battery is fully charged and inserted correctly. Check your camera's manual to see if there is something else specific to your model.
Some models have a second battery (or high capacity capacitor?) inside. Its purpose is to maintain the clock and setup memory while the battery is out of the camera for charging. Over time, the camera's battery self-discharges, and this internal battery will also run down. In case that has happened to your G1, it may self-correct if you keep a fully charged battery in it for several hours.
One Canon EOS DSLR had a separate lithium cell inside for the same purpose. It should last for several years, but eventually needs replacing.