Wash and chop the cucumber. Puree very finely with syrup in a blender or with a hand blender. Pass through a sieve, collecting the cucumber syrup.
Pour about 200 ml of syrup and crushed ice into four glasses. Top up with wine and mineral water. Decorate with mint and serve immediately. If cucumber syrup is left over, it can be kept in the refrigerator for a maximum of 3 days.