Sounds like the battery is dead or has kicked the bucket. (Time for a new battery)
Yes, Dell batteries aren't very good. I'm not impressed with Dell. They also fall apart easily. I have an HP which i'm reasonably happy with.
Wasn't Dell one of the occupancies who had a major battery recall?
If the AC adapter keeps it powered on it's probably the battery. If you want to save the laptop look for a new battery online, lots of places sell both old and new model replacements.