Bring sugar and the water to a simmer in a small pan, stirring. Remove from heat; whisk in cocoa. Transfer syrup to metal bowl; chill, about 40 minutes.
Puree berries, lemon juice, and syrup in blender in 2 batches until smooth. Strain through a fine sieve, discarding seeds. Freeze in an ice-cream maker following manufacturer's instructions. Transfer to another container, cover with plastic wrap, and freeze fully.