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.