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 remove 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 chill completely in the refrigerator.
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.