Prepare the simple syrup: Place the sugar and 4 cups cold water in a saucepan and bring to a boil over high heat, stirring with wooden spoon until the sugar dissolves. Continue to boil for 5 minutes and remove from the heat. Cool and refrigerate in a covered container. Simple syrup keeps for 3 to 4 weeks refrigerated. In a large pitcher, combine the remaining lemon juice and simple syrup and 3 cups soda water and mix. Pour over the lemonade ice cubes. Add a sprig of mint and straw to each drink. Serves 6.