Puree blackberries in a blender until smooth; strain into a bowl to remove seeds. In a small saucepan, heat sugar and 1 cup water until the sugar is dissolved. Pour mixture into the strained berries, and stir in lemon juice and another cup cold water.
Pour into a 9" x 13" glass dish, and place uncovered in the freezer. Stir with a fork every 20 to 30 minutes until frozen, about 2 hours.
When ready to serve, use an electric mixer to beat cream until soft peaks form. Garnish each granita with a dollop of whipped cream and whole berries.