Heat the blackberries, sugar and 1 tablespoon water in a small pot over medium heat, smashing the berries with a potato masher, until thick, about 5 minutes. Pour the mixture through a sieve into a bowl, pushing the juice through with a rubber spatula. Refrigerate until cold, at least 10 minutes.
Fill a cocktail shaker with 2 tablespoons of the blackberry juice, 2 tablespoons mint, 2 ounces gin, 1 tablespoon lime juice and a handful of ice. Shake vigorously until the drink is ice cold, 5 or 6 times. Strain into a tall glass. Top with ginger ale and stir lightly. Repeat to make 3 more drinks. Top each with a scoop of ice cream.