Thaw the berries, if frozen. Do not drain. Place half of the berries in a blender container or food processor bowl. Cover and blend or process until berries are smooth. If using raspberries, press berries through a fine-mesh sieve; discard seeds. Repeat with remaining berries. (You should have about 1/2 cup puree from each 1-1/2 cups berries.)
In a medium saucepan stir together sugar and cornstarch. Add berry puree. Cook and stir over medium heat until thickened and bubbly. Cook and stir for 2 minutes more. Remove from heat. Cool to room temperature before serving. Makes about 1 cup sauce (sixteen 1-tablespoon servings).