Wash basil, shake dry, pluck leaves from the stalks. Wash the cucumber thoroughly. Peel all around with a peeler. Set aside strips of peel for decoration.
Chop the rest of the cucumber roughly. Add basil and sugar. Puree finely with a hand blender. Pass through a sieve into a large carafe. Squeeze limes. Stir 125 ml juice and gin into the carafe.
Spread crushed ice and cucumber peel strips in glasses. Add cucumber gimlet. Top up with mineral water.