Sprinkle gelatin over 1 cup cold Prosecco in a large bowl. Let stand 3 minutes.
Stir together sugar and remaining 2 cups Prosecco in a medium saucepan. Bring to a boil over high heat, stirring constantly; boil, stirring constantly, until sugar dissolves. Add hot Prosecco to gelatin mixture, and stir until gelatin dissolves. Chill until consistency of unbeaten egg whites (about 40 minutes).
Place raspberries and blackberries in a 6-cup gelatin mold. Stir cold club soda into chilled gelatin mixture until blended. Pour over fruit in mold. Chill until set (about 2 hours).
If necessary, run a small knife around edge of mold to break seal. Dip bottom of mold in warm tap water for about 15 seconds. Unmold onto a serving plate.