It has been bitterly (polar vortex) cold around here. I thought that we were good on the plumbing front, but today as temps have started to rise (it's 1 degree F now), I have a water faucet that is running without being turned on. I cannot get it off. It is in the coldest part of my house. All other faucets are working fine.

It would be easier to assist you with your issue if you could tell me what type of faucet it is. Also is there a shut off directly under or away from that faucet that controls it?

Depending on the type of faucet it is possible that it was frozen and from attempting to turn it it it broke. It has happened to me in NY before.