In a blender, puree the blackberries with 1 cup of the water. Strain through a fine sieve. In a small saucepan, combine the sugar with 1 cup of the water and bring to a boil. Simmer until reduced to 1 cup, about 15 minutes; let cool.
In a pitcher, combine the blackberry puree with the sugar syrup, lime juice and the remaining 5 cups of cold water and refrigerate until chilled. Serve in tall glasses over ice.