Heat the sugar and water together in a small saucepan over medium-low heat; cook and stir until the sugar is dissolved completely, making a simple syrup. Set aside to cool.
Puree the blackberries in a blender or food processor; strain through a fine-mesh strainer to remove any seeds or pulp. Whisk the strained blackberry juice, the cooled simple syrup and the red wine together in a bowl; chill in refrigerator until cold.
Freeze in the freezer canister of an ice cream maker according to the manufacturer's directions.