Bring sugar and 1 cup water to a boil in a small saucepan, stirring constantly. Boil, stirring constantly, 3 minutes or until sugar dissolves. Remove from heat; stir in mint. Let stand 5 minutes. Pour through a fine wire-mesh strainer into a large bowl; discard mint. Stir in berry juice.
Pour mixture into a 13- x 9-inch pan; freeze at least 8 hours. Let stand at room temperature 20 minutes. Process, in batches, in a food processor. Serve immediately.