Lime Ice on Watermelon

1 1/4 cups lime sugar
1 1/2 cups granulated sugar
2 1/4 cups water
4 whole star anise (optional)
1 3/4 cups fresh lime juice (10 limes)
1/2 chilled watermelon, cut into slices or wedges

Boil sugars, water, and star anise, stirring, until sugars are dissolved. Pour syrup through a fine sieve into a bowl, discarding solids. Cool syrup.
Stir in lime juice.
Freeze in a 13- by 9- by 2-inch glass dish, stirring occasionally with a fork, until liquid is frozen and granular, about 6 hours.
Serve lime ice on watermelon.

Cooks' note:
Lime ice may be made 2 days ahead and kept frozen, covered.