How to tell if battery is dead

I am trying to fix my 1st gen iPhone that I just bought as non working. It seems that the battery is bad, but I want to make sure that it's really the problem before ordering one. Whenever I tried to restore it in itunes or 3utools, it always tried to reboot and just turned the screen off every time. Sometimes, the backlight would turn off and a ghost of the screen could be seen for a few seconds. I can get the phone in DFU and recovery mode no problem, but as soon as I unplug it, it shuts off immediately. With the latest itunes I would get error 1604 and 1603. Is this likely the battery or is the dock connector or motherboard bad? I tried to download older itunes versions, but even on vista or Windows 7 machines I had no luck showing the “iphone software update server could not be contacted”.