Process 1/2 cup of the Blackberries in a food processer or blender until pureed. Combine the cornstarch and sugar replacement in a heavy or nonstick saucepan; then gradually add the milk, pureed Blackberries, and lemon juice.
Cook and stir over medium heat until clear and thickened. Remove from heat and stir in remaining Blackberries. Cover and cool to room temperature before using.