In a medium saucepan, combine blackberries, 2 tbsp. lime juice, and 1 tbsp. granulated sugar.  Cook over medium heat for 10 minutes or until berries start to break down.  Strain cooked blackberries through a sieve to remove seeds and pulp centers into a bowl.  Discard seeds.  Return seedless blackberries back to saucepan.  Stir in remaining 3 tbsp. granulated sugar.  Bring to a boil over medium heat.  Combine cornstarch and cold water in a small bowl.  Stir into boiling blackberry mixture and cook until filling is thick, stirring constantly.  Allow blackberry filling to cool completely before using.*