Place the blueberries
and the water into a medium saucepan, set over medium-high heat and bring to a boil. Reduce the heat to low and simmer
for 15 minutes. Remove the saucepan from the heat and pour the mixture into a colander
lined with cheesecloth that is set in a large bowl. Allow to cool for 15 minutes. Gather up the edges of the cheesecloth
and squeeze out as much of the liquid as possible. Discard the skin and pulp. Return the blueberry
juice to the saucepan
along with the sugar and lime
juice. Place over medium high heat and stir until the sugar has dissolved. Bring to a boil and cook for 2 minutes. Remove from the heat and transfer to a heatproof glass container and place in the refrigerator, uncovered, until completely cooled.
To Serve: Combine 1/4 cup of the liquid with 8-ounces of carbonated water and serve over ice.