Cut two thin slices from the cucumber for garnish. Peel the remaining cucumber and halve lengthwise. Remove the seeds and coarsely chop.
Puree the cucumber with the dill, lime juice, and a few drops of Tabasco sauce in the blender. Season to taste with salt and pepper. Pour the puree into a cocktail glass, then top with the mineral water. Garnish the glass with the cucumber slices, and serve.