Put the blackberries in another pan. Squeeze some lemon juice over them and simmer over low heat for a minute or two until soft.
Put the blackberries in a blender and purée until smooth.
Push the purée through a fine sieve to remove all the seeds.
Add the cooled syrup and remaining water to the purée and whisk together.
Pour the mixture into an ice cream maker. Process for about 20 minutes according to the manufacturer's instructions, then freeze.