Combine coffee, sugar, and marshmallows in a 1 1/2-quart saucepan; cook over medium heat, stirring constantly, until marshmallows melt and mixture is well blended. Chill.
Combine coffee mixture, whipping cream, half-and-half, vanilla, and salt in a large mixing bowl, stirring until well blended. Pour mixture into freezer can of a 1-gallon hand turned or electric freezer. Freeze according to manufacturer's instructions. Let ripen 1 1/2 to 2 hours before serving. Scoop into individual bowls, and serve immediately.