Add blackberries, water and sugar to a medium saucepan and cook on low until the berries are soft (10 – 15 minutes). Stir occasionally with a wooden spoon, slightly mashing the berries each time you stir.
Pour the mixture through a fine mesh basket strainer then return to the saucepan. Discard solids. Add balsamic vinegar and rosemary sprigs. Increase the heat to medium-low and bring the mixture to a slow boil.**
Take the mixture off of the heat and carefully remove the rosemary sprigs. Let cool then transfer to a pitcher or bottle. Chill shrub in the refrigerator before using.