Look for the ripest melon you can find; it should have a sweet fragrance and a firm exterior that yields slightly at the stem end. We like to serve the granita over wedges of cantaloupe for added melon flavor.
Using a sharp knife, cut melon in half lengthwise. Remove seeds with a spoon, and discard. Slice off and discard skin and pale-green flesh. Cut melon into large chunks.
Place in the bowl of a food processor fitted with the metal blade, and puree until smooth. Transfer to a nonreactive stainless steel bowl, and set aside.
Combine sugar, lemon juice, and the water in a small saucepan. Cook over medium heat until mixture has thickened slightly. Remove from heat; let cool completely.
Stir sugar syrup into melon puree; place in freezer, uncovered, until mixture is chunky, about 1 1/2 hours, whisking every 20 minutes to keep it from becoming too solid. To serve, divide among four shallow bowls.
Granita can be stored in an airtight container in the freezer for up to 2 weeks.