I have a Dell Inspiron E1505. I am running Linux Mint 16 but it did the same thing with another distro.
When it says the battery is fully charged and I remove the power cord the battery runs for a short time (a few minutes) and then dischages almost immediately. I get a red battery outline and I must plug in the charger cord to keep running. Then it takes quite awhile to fully charge the battery.
What's causing the rapid discharge? Any ideas.