Starter keeps running after key in run or off

I have a '66 Impala, and just put a reman starter on it. When I turn the key to the "run" position, the dash lights come on as they should; when I turn the key to "start" position, the starter engages as it should... BUT, when I release the key to "run", or "off" positions, the starter motor remains engaged and running. I have dealt with a lot of starters in my time, but this one has me stumped.

i would have to guess the starter solenoid .

I agree. I had that happen many years ago on an Olds. It's very unnerving. Had to pull the battery cable -- be careful, pull the ground. I pulled the hot and it was a shocking experience.