Rinse the blackberries, then combine in a blender with the water and lemon juice. Purée until smooth, then press the mixture through a sieve to grab the seeds.

In a saucepan, whisk together the blackberry juice and sugar. Bring to a boil and reduce to a simmer. Let cook for 1 to 2 minutes, until the sugar is fully dissolved. Remove from the heat, transfer to a container, and place in the refrigerator to chill completely.

Freeze the sorbet mixture according to your ice cream maker’s directions, or, if not using an ice cream maker, place the mixture in a shallow pan and freeze. Once frozen, break into pieces and purée in a blender or food processor until creamy.