Peel melon, remove seeds, cut into small cubes and machine-blend. Keep cool. Slice ginger finely and boil half of it in water with half the sugar until sugar has dissolved. Turn heat down, and add noodles to simmer for 5 minutes. Remove from heat, allow to cool, pour into a bowl, and remove ginger and chili.
In a saucepan, boil together remaining sugar, ginger and lime or lemon juice. Simmer until thick. Remove from heat, cool and remove ginger. Chill.
In individual bowls set in larger ice-filled bowls, pour equal quantities of gingered melon puree. Top with noodles then lime or lemon mix. Garnish each with a mint leaf and serve.
