It can be the battery.

But how do you find out if it is?

You can buy a new battery and find out.
But that means spending $15 - $20 just to find out.
If it turns out to be the camera, you are out that money - which you could have put toward a new camera.
If it was the battery - you have a working camera again.

If you have a friend who is into electronics, that friend may have a volt-meter. They could measure the output of the battery and it should read 3.7 volts.

That camera uses a NP-80 battery which is quite common and can be obtained on-line.
Just do a Google search for --> battery np-80