In a blender or food processor, blackberries, sugar, lemon juice, and salt until smooth.
Press through a fine strainer into a bowl; discard seeds. Taste, and add more sugar (up to 1/4 cup) if desired.
Stir in half-and-half. Cover and chill until cold, about 1 hour.
Pour into an ice cream maker (1 1/2-qt. or larger capacity); freeze according to manufacturer's directions until mixture is softly frozen. Serve soft, or freeze airtight until firm, about 3 hours, or up to 3 days.