Engine problems when it is cold outside

I don't know if the above is accurate - live in Michigan car quits when freezing and snowy weather. Does fine when mild weather

Car will quit after about 5 minutes on the road but only when it is freezing and snowing. Let it wait for 10 minutes or so it will start again, but quit a minute after moving. My genius mechanic had it when it was mild weather though and obviously could not reproduce the problem. He said to keep 1/2 of a tank of gas or more and that should do it, it has not. I am sure lots is rusted underneath as the gas and break lines seemed to be metal.