Craftsman Riding Mower Won't Start?

My 1997 Craftsman 19.5 lawn mower will not turn over with the ignition. When I try starting the tractor, it makes one clicking sound and the amp meter moves. I thought it was the battery at first and tried to jump it, this did nothing. I changed out the battery with another that was working fine for another tractor that I have, did nothing as well. I determined that the battery is fine and then I ran a jumper wire from the positive terminal to the starter and the tractor turned over and started like normal.

I jumped the solenoid with a screwdriver and this made the starter turn over. I believe that the solenoid is bad and needs to be replaced. Am I correct in assuming this or is there a switch between the ignition and the solenoid that could be bad? Or could the ignition switch be bad? I don't think this is the case because the ignition switch is causing the amp meter to function.

By Brent B.