Wash organic lime in hot water, dab dry and rub the skin. Squeeze all limes and measure 1⁄8 l juice. For the sugar syrup, boil 300 ml water, sugar and lime zest while stirring until the sugar is dissolved.
Let it cool down.
Beat egg whites until stiff. Stir in the sugar syrup and lime juice. Pour into a bowl, place in the freezer and let it freeze for about 1 hour. Then stir well and freeze for another 2 hours.
Stir vigorously every 30 minutes (preferably with a whisk).
Cut 4 large balls from the sorbet mass with an ice-ball scoop and divide them between four glasses. Fill up with gin. Decorate with lemon balm and serve immediately.