Remove the rind from the watermelon and discard. Chop the flesh into 1-1/2-inch pieces; you should have about 5 cups of fruit. Purée the watermelon in a blender, in batches if necessary, until smooth. Strain through a fine sieve and discard the pulp; you should have 4 cups of juice. Add the sugar and lime juice to the watermelon juice. Stir with a large spoon or whisk until the sugar has thoroughly dissolved.
Pour the base mixture from one of the recipes at right into a 9-inch-square shallow baking pan. This pan size works best because it provides a large surface area, a key point in speeding up the freezing process. To further hasten freezing, use a metal pan (metal conducts cold well).
Put the pan in the freezer and stir every 30 minutes, being sure to scrape the ice crystals off the sides and into the middle of the pan, until the mixture is too frozen to stir, about 3 hours, depending on the individual recipe and on how cold your freezer is (some granitas can freeze in as little as 1 hour). Use a large dinner fork to stir and scrape; the tines are perfect for breaking up ice crystals.
Cover the pan with plastic and freeze overnight. When ready to serve the granita, place a fork at the top of the dish and pull it toward you in rowsm moving from left to right and rotating the pan as well. Scape up the shaved ice and fill your chilled glasses or bowls.