battery fully charged, but computer won't turn on with it

Hi,
I have a 14 inch ibook g4 that's about 3.5 years old and has the original battery which has been getting about an hour of life. The other day I unplugged the AC cord and the computer immediately turned off. When I reattached AC power, it would not turn back on. But it works with AC power if the battery is removed (which is what I am currently doing). It still does not work with the battery attached even though all 4 lights are lit when I press the back button, so i don't think it's completely dead. Do I just need a new battery or is it a connection problem? Thanks.