i run unbuntu - there is no windows partition on this machine

my netbook just powers off and will not come back on. i thought it was the AC adapter, got a new one and it worked for a while. when the problem returned, i checked the adapter for continuity, and they had it. i took the sucker apart to check the DC in and found it also had continuity. when i put it back together it booted and work for a bit, showing no battery charge (0%) then shut off and now won't come back on. i tried chilling it, no dice. tapping on the fan, no dice.

one) should the netbook work with the battery removed and the AC adapter plugged in?

two)how can i test the battery?

i *really* need for this to be fixable - any idea would be helpful.

thanks.