Grate 1 tablespoon peel (yellow part only) from about three lemons. Ream juice from enough lemons to make 3/4 cup. In an 8- or 9-inch square baking pan, mix peel, juice, and 1 1/2 cups water.
In a food processor or blender, whirl sugar until very fine. Add basil and whirl until very finely minced. Stir sugar mixture into lemon mixture until sugar is dissolved.
Cover and freeze mixture until firm, about 6 hours.
Scrape the tines of a fork quickly across frozen mixture to break into fluffy granules. Scoop granita into chilled bowls or wine or sherbet glasses and serve immediately.