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.
Perfectly peachy keen! Almost too sweet, but it works. You can barely taste the rum, so add more if you want more of a cocktail. I would serve right after processing until slushy, instead of freezing another 3 hours next time.
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