In a 1 1/2- to 2-quart pan over high heat, frequently stir blueberries, 2/3 cup sugar, and 1/4 cup water until berries begin to pop, 4 to 5 minutes.
In a blender or food processor, pure blueberry mixture, raspberries, and 1 tablespoon lemon juice. Rub through a fine strainer into a bowl; discard residue.
Add more sugar and lemon juice if desired. Nest bowl in ice water; stir often until mixture is cold, 15 to 20 minutes.
Pour chilled pure into an ice cream maker (1-qt. or larger capacity). Freeze according to manufacturer's directions until sorbet is firm enough to scoop, the dasher is hard to turn, or the machine stops.
Serve, or freeze airtight (see below).
To get frozen desserts hard enough to scoop onto a cone, or to store them, transfer when frozen to an airtight container and put in the freezer at least 3 hours or up to 1 week.
If freezing with ice and salt, leave the frozen dessert in ice and salt up to 3 hours.
For best flavor and texture, serve frozen desserts within a week. On longer standing, icy crystals develop.
