Odd problem starting up Powerbook 17"

I have a Powerbook 17" 1.33 GHz that has developed an odd problem. As the battery is no longer able to hold a charge for any length of time I leave the Powerbook on all the time with the power cable plugged in.

For some reason that I don't understand the power cable will stop supplying a charge (the light on the end of the cable goes off) and the Powerbook shuts down. If I unplug the power cable and re-plug it in the light on the cable end shows but the Powerbook won't start. It is completely dead when I press the start button, no response at all.

If I leave it with the battery and power cable removed overnight it will usually start the next morning and I get a message saying that the clock is set to the wrong date and everything is fine until the power gets cut off again.

Can anyone tell me what is going on with the laptop and do you think I need a new power cable or is the problem with the computer?

Thanks