[Selectron]

Assuming that the wire which you're unhooking at the starter solenoid is the smallest of the three wires - should be black/red, and that's the one which connects to the small 'S' terminal and energises the solenoid - then I'd agree that it's probably the ignition switch. That wire shouldn't receive 12V until the ignition switch is turned to the Start position, so if it's already receiving 12V in the On position then likely the ignition switch has failed internally.

There's a remote possibility that the black/red wire is shorted out to some other wire which receives 12V in the On position, but that really is a long shot. If you replace the switch it will probably cure it. As I'm sure you know, you don't need a lock cylinder - just a new switch section.