It could be either one Sarah.
The AC adapter's (Charger) cable, or the DC Power Jack where the AC adapter plugs into the laptop.
This is an example of the DC Power Jack not installed,http://www.laptopjacks.com/view_part/Acer-Aspire-ONE-AOA150-ZG5-DC-Power-Jack-AC351816.html
(You can click on the three views of the jack to enlarge)
1) Check the DC Power Jack to see if it's in good repair.
Battery removed use a small flat tipped screwdriver, and Gently see if you can move the Center Pin around in the jack.
See if the Center Pin will move side to side a small amount. There is supposed to be No perceptible movement.
Referring back to the closeup photos of the DC Power Jack, the flat blade-like prongs are upside down, in relation to how the jack sits on the motherboard.
The flat prongs go down through the motherboard, and are soldered to the motherboard.
If the center pin of the jack is stationary, but the jack as a unit moves around, one or more solder joints, (Connections) for the flat prongs may be cracked.
The repair consists of completely disassembling the laptop, removing the broken DC Power Jack, and installing a new one.
Acer Aspire One service manual,http://www.tim.id.au/blog/tims-laptop-service-manuals/#toc-acer
(Click on the blue - One - in Aspire One.
This is a PDF file. The computer you are using now has Adobe Reader on it which uses PDF files.
After you click on the file name (One) it may take up to 30 seconds before you see the first page.
It may also take additional time for the file to fully download.
(Took 12 seconds just now for full download)
No perceptible movement of the DC Power Jack? Suspect the cable of the AC adapter right near it's plug. (Plug that plugs into the DC Power Jack)
For additional questions please post in a Comment.