Combine chopped peaches and 1/2 cup water in a blender or food processor; process until smooth. Press peach mixture through a fine sieve into a bowl; discard solids.
Combine 1 cup water and sugar in a medium saucepan; bring to a boil.
Remove from heat. Stir in peach mixture, juice, and rum.
Pour mixture into an 8-inch square baking dish; cover and freeze 4 hours or until firm.
Place mixture in a food processor; process until slushy. Freeze 3 hours. Soften slightly in the refrigerator for 30 minutes before serving.
