In a medium saucepan, combine the sugar, water, blackberries, and lime zest.
Bring to a simmer, stirring occasionally to dissolve the sugar and pressing the blackberries against the side of the pan with the back of a wooden spoon.
At the first crack of a boil, remove from the heat. Once cooled, pour the syrup through a strainer, discard the solids, and store in a glass jar.