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Put the cucumber and half the mint in a large bowl or jug, and pour over the vodka. Cover and leave to steep for at least 24 hrs or for up to 1 week in the fridge, stirring every few days. Strain through a fine sieve.
Pour the vodka, lime juice and sugar into a larger jug. Stir well.
Using a melon baller, scoop out as many melon balls as you can. Add them to the jug and leave to steep for 30 mins-1 hr in the fridge. Top up with sparkling water and add the remaining mint leaves just before serving with extra lime wedges.