Place mango flesh in food processor bowl with metal blade. Add juice cocktail concentrate; process until smooth. Pour into 2-quart nonmetal freezer container. Stir in water and rum. Cover; freeze 6 to 8 hours or until icy, stirring twice.
To serve, spoon 1/2 cup frozen mixture into each glass. Add 1/2 cup carbonated beverage to each glass; stir gently.