Sponsored Topics. Mount Vernon is a town in Franklin County, Texas, United States. It is the Franklin County seat, and the population was 2,286 at the 2000 census. Mount Vernon is located at 33°10′55″N 95°13′27″W / 33.18194°N 95.22417°W / 33.18194; -95.22417 (33.181890, -95.224147).