After reading your replies, I think we are having some SERIOUS mis communications here..
When you say switch..Are you talking about the place where the key goes in? OR are you talking about the solenoid???
The solenoid will have two large cables on it and two smaller wires on the front marked "S" and "I'.. = SOLENOID..
If you reversed the "S" and "I" wires, the starter will turn the engine anytime the key is on because the primary Ignition is back feeding the "s" terminal where the "I" wire was mistakenly put.
When you say the starter just runs all the time, do you mean without engine running or with it running? You said you had to remove the ground cable to shut it off..the engine or the starter?
The alternator wire goes to the battery side of the solenoid (big bolt) and should not effect run on if everything else is properly hooked up..