Cut the melon pulp from the skin, dice roughly, removing any seeds. Finely puree the melon with the chopping stick, place in the hollows of an ice cube maker and freeze for at least 2 hours.
Wash the lemon balm and dab dry. Spread melon ice cubes on 4 long drink glasses. Put a stem of lemon balm into each glass, pour fresh, cold drinking water over it and serve. Refreeze the remaining ice cubes and use them for other purposes.