I would use strong tie connectors around the stud you are concerned about. The flat ones you can screw in on the sides on the stud. And then cut the mold out and spray what remains with bleach. That way you will 100% be set. You will likely be set doing nothing but this will give you complete confidence in what remains.

A previous post talked about a foam cover and if you are in the home this is the best option ( aside from a frost proof valve ) if you are not there and using water this method is useless.

They look like a foam breast and attach to the valve with rubber bands and seal against the house. Or they have a string you tie around the faucet and then you cinch it up against the front of the foam. They work perfect and I have used many times for friends when I was unable to help them install a frost proof valve, which again is the ultimate solution.

Worst case and should be avoided is to wrap blankets/insulation around the valve with duct tape and let it drip, which usually prevents freezing.