In the bottom of a tall pitcher, use a wooden spoon to gently mash the mint leaves with 1/4 cup of the sugar. Add the strained juice; stir well to dissolve the sugar. Add the cold water and the remaining 1/2 cup sugar to the pitcher; stir well to dissolve the sugar. Taste for sweetness and add more sugar, if desired. Chill before serving.
Serve poured over ice in tall glasses garnished with lemon and lime slices.