Combine 2 cups water, sugar, and chopped mint in a small saucepan; bring to a boil. Cook until sugar dissolves, stirring frequently. Remove from heat; let stand 10 minutes. Strain through a sieve into a bowl; discard solids.
Combine the remaining 4 cups water, sugar syrup, and lime juice in a large pitcher, stirring well. Serve over ice; garnish with mint sprigs and lime slices, if desired.