Acer Aspire one Battery Question

I purchased a reconditioned Asphire One Netbook and recently noticed that the Battery is not charging???? Once I unplug the power source,the computer shuts down. Common sense tells me I have a bad (Dead) battery, but I wanted to check with you before ordering a new one. Could there be any other reason the battery is not holding charge??? Thank You in advance for any answers, and forgive me if this subject has been posted at an earlier time. I did a quick scan before posting..

 

This is common. I buy lots of the refurbished ones for education and many fail to charge for a start. Strangely enough I find that if you charge it all night it will still appear flat when tried on battery in the morning. However, if you plug it in and leave it for a while booted up, then maybe charge again, it will eventually start to charge normally and perform off the battery.