Scoop cantaloupe, watermelon, and honeydew with melon scoop or spoon. Pile into serving bowl with blueberries and chill.
Mix sugar and water; bring to boil. Simmer for 5 minutes. Add lime rind; cool to room temperature. Stir in lime juice and rum. Pour lime sauce over melon balls and berries and chill, covered, for several hours.
Decorate with mint sprigs. Makes 8 servings. Lovely, refreshing hot weather dessert.