Combine the cranberry juice and sugar in a medium saucepan. Heat and stir until sugar is dissolved. Add the 1 cup lemon balm leaves to saucepan. Bring to boiling; reduce heat. Cover and simmer for 10 minutes. Strain into a large mixing bowl; discard lemon balm. Stir in sliced strawberries. Cool mixture to room temperature.
Place half of the strawberry mixture in a blender container or food processor bowl. Cover and blend or process until smooth. Transfer mixture to a nonmetal freezer container. Blend or process remaining mixture; add it to the freezer container.
Cover and freeze mixture about 4 hours or until almost firm. Break mixture into small chunks; transfer to a chilled bowl. Beat with an electric mixer until smooth but not melted. Return to container. Cover and freeze for 6 hours or until firm.
To serve, scoop mixture into individual dessert dishes. If desired, garnish each serving with lemon balm leaves. Makes about 6 cups or 12 servings.