Place chopped mint on a double layer of cheesecloth. Gather edges of cheesecloth together, and tie securely.
Place cheesecloth bag and apple juice in a large saucepan; bring to a boil. Remove from heat; cover and let cool. Pour into a large pitcher; cover and chill. Discard cheesecloth bag, and stir in lime juice.
Pour apple juice mixture over ice cubes. Garnish with additional mint sprigs, if desired.
Note: To make Apple Julep Granita, pour apple juice mixture into a 13- x 9-inch baking dish. Cover and freeze until firm. Remove mixture from freezer; scrape entire mixture with the tines of a fork until fluffy. For an extra-special beverage, make apple julep ice cubes. Pour some of the apple julep mixture into ice cube trays, place a mint sprig in each and freeze. Serve with apple julep mixture or with sparkling water.