Laptop unplugs when AC adaptor is unplugged.

I own a Sony Vaio SVF14AC1QL, after about a year of me having it the laptop would turn off immediately when unplugged. The battery said something along the lines of "0%, not charging." I ordered a new battery and changed it out, and now the battery says "70%" but won't go any higher. The bigger problem is that it still turns off immediately when the charger is unplugged. 

I thought about ordering a new charger or ANOTHER battery, but I wanted some input before wasting money. Is this a problem with the battery, the charger, or the laptop itself? Any ideas how I can fix it?