It could be a dead/dying battery, a problem with the AC adapter, the connections between the battery and the charger, the battery is not properly attached, or a number of other things.
The first thing I would do is remove the battery and replace it securely to make sure it is properly installed and see if that makes any difference.
Go to http://support.acer.com/acerpanam/manuals/acer/0000/userguides/AS6920%20user's%20manual%2020080219.pdf
and start at page 43 for information about the battery.
Yes, you can use the computer with just the AC adapter and no battery (in fact, considering how you say you use the system, you should be doing this much more frequently to prolong the battery as you'll read in the above manual - and yes you should take the battery out because you can't just turn it off and this will reduce the chances of causing problems with the adapter trying in vain to charge it), but bear in mind that in the event of a power failure, you will experience a hard shutdown as opposed to going over to the battery (which is not good for the system), so if you intend to do this for prolonged periods, I recommend a UPS battery backup system to give you time to shutdown correctly (and protect the system from surges).
It's hard to tell what the problem is without having replacement parts to test which component is causing the problem and opening the case to check all the connections (not recommended on a laptop unless you are experienced at such things, have the right tools, and know what you're doing and looking for). I suspect it is the battery because of how you are using the system with it constantly plugged in and never actually using or draining and recharging the battery. Batteries typically have a different warranty than the rest of the computer - usually only one year at most - but Acer can tell you that: http://us.acer.com/acer/contact_us.d...CRC=3529792491
It doesn't matter much where you get a replacement battery as long as it is EXACTLY what fits the make and model of your laptop. Acer is pretty sure to get that right. Anywhere else and you'll need to verify it. The wrong battery could harm your system.
I hope this helps.