Combine orange juice and sugar, and stir until sugar dissolves. Transfer to an 8-inch square baking dish. Freeze, scraping around sides and breaking any lumps with a fork every 30 minutes, until evenly frozen, 2 1/2 to 3 hours. Cover with plastic wrap. Granita can be made 1 day ahead and kept in freezer until ready to serve.
Spoon 1/2 cup granita into each of 12 glasses, and add 1/2 teaspoon liqueur. Top off with Champagne.