In a small saucepan, combine the sugar with 1/3 cup of water and bring to boil over high heat, stirring, until sugar dissolves. Transfer the syrup to a blender or food processor, add half of the watermelon cubes and the lime juice and pulse until smooth. Add the remaining watermelon cubes and blend until smooth.

Pass the puree thorugh a coarse strainer, pressing down on the solids. Transfer to a 9x13 inch nonreactive baking dish and freeze, stirring every 30 mintues with a fork, until all the liquid has frozen completely, about 3 hours. (Can be made ahead to this point and frozen up to 2 days) Spoon the granita into tall glasses or bowls and garnish with the melon wedges.