Bring the water, sugar, and half of the mint to a boil. Cook for 5 minutes, then trasnfer to a carafe. Add the lemon slices to the carafe.
Chill in the refrigerator for several hours. Add the lemon juice and the mint to the carafe, and fill with the mineral water. Chill for 1 more hour, then sweeten to taste with a little sugar or another sweetener. Serve the lemonade well chilled.