Wash the lime and the mint under cold water. Roll the lime on a wooden board with some pressure. Then cut the lime into quarters and put into a cocktail glass (with a heavy base).
Add the cane sugar and crush with a wooden pestle. Add 3 to 4 mint leaves and crush as well (carefully, so the mint aroma can develop).
Add the rum, then fill half of the glass with crushed ice or ice cubes (preferably crushed ice; use the stamper to crush it). Stir, then add a dash of soda water (or mineral water with bubbles).