Place sugar, tarragon, apple juice, and lemon juice in a blender and blend until well combined and tarragon is finely chopped. Let mixture stand for 10 minutes and then pour into a 9-inch square metal baking pan. Freeze the mixture, stirring and crushing lumps with a fork every hour, until evenly frozen, about 4 hours.
To serve, scrape with a spoon into chilled dishes or martini glasses. Garnish with fresh tarragon sprigs.