Puree berries and 1/3 cup water in a blender. Pour through a coarse sieve into a bowl. If using blueberries, strain again, through a fine sieve. (You should have 2 cups.) Stir in simple syrup.
Refrigerate until cold, about 1 hour. Process in an ice cream maker according to manufacturer's instructions, about 25 minutes. Transfer sorbet to an airtight container, and freeze until it hardens, at least 1 hour and up to 3 days.
Currant sorbet: Use 1 1/2 pounds fresh red or black currants, stems removed (4 1/4 cups). Bring currants and 1/2 cup water to a boil. Reduce heat, cover, and simmer until berries are soft, about 5 minutes. Puree mixture (do not add 1/3 cup water), and strain as above. Proceed using 1 to 1 1/4 cups simple syrup for red currants and 1 1/2 cups simple syrup for black.