The day before, for the ginger syrup, bring the sugar and water to the boil in a small pot and add the ginger, cut into fine slices. Simmer over a low heat for ten minutes. Cool, pour into a screw-top jar and leave to stand overnight.
Peel the melon, remove the stones and dice the pulp. Mix with two tablespoons of ginger syrup, without pieces of ginger, and the lime juice and refrigerate for about 30 minutes. Add the mint leaves and season to taste.