1. With a small, sharp knife, peel the mangoes, cut the flesh away from their stones, and chop it coarsely. Rub the mangoes through a fine sieve set over a bowl, pressing down hard with the back of a large spoon before discarding the pulp. (Or puree them in a food mill.) Stir the lime juice into the puree and set aside.
2. In a small, heavy saucepan, bring the sugar and water to a boil over moderate heat, stirring until the sugar dissolves. Then cook briskly, undisturbed, for about 5 minutes, until the mixture becomes light syrup. Pour the syrup into a heatproof mixing bowl and let it cool to room temperature.
3. Stir the mango puree into the cooled syrup and pour the mixture into an empty ice-cube tray. Freeze for 3 or 4 hours, stirring the mango ice every 30 minutes to break up the icy particles as they form.
4. The finished mango ice should have a fine, snowy texture. To serve, spoon the ice into chilled parfait glasses or individual dessert dishes.