In a pot, bring the sugar and water to a boil. Pour in the lime juice and simmer over medium heat, stirring occasionally, for 15 minutes. Cool, pour into a clean, dry bottle, and cork tightly.
In a pitcher, mix 1 part syrup, 3 parts water, and 2 ice cubes per person. Stir with a spoon and serve well chilled. Garnish with sprigs of fresh mint and slices of lime.