In a large pot, boil 2 cups of water with the sugar. Once dissolved, remove from heat and allow to cool.
Juice the lemons. In a blender, mix the lemon juice and blackberries and blend. Strain the mixture and discard the seeds.
In a pitcher, stir together sugar water, the blend from the second step, and add 4 additional cups water. Refrigerate until cool or add ice cubes to individual glasses. Serve with a wedge of lemon.