Have ready a large bowl of ice and cold water. Bring a 6-quart pot three-fourths full of water to a boil, then gently drop peaches into water and cook 15 seconds.
Transfer with a slotted spoon to ice water and cool completely to stop cooking.
Remove skin with a paring knife and cut peaches into 1/2-inch-thick wedges, discarding pits.
Melt butter in a 10- to 12-inch heavy skillet over moderate heat, then stir in brown sugar, rum, cinnamon, and salt until smooth.
Add peaches and simmer, stirring and turning peaches over occasionally, until tender, 3 to 5 minutes. Cool until warm and serve over scoops of ice cream.
Peaches in sauce can be made 2 hours ahead. Reheat to warm before serving.
