Pour 100ml water in a small saucepan and bring to the boil. Remove from the heat, tip in the sugar and stir until dissolved. Add the mint leaves, return to the heat and simmer for 5 mins then remove and leave to infuse for 30 mins. (This can be made the day before and left in the fridge overnight to infuse).
Pour through a sieve into a jug, squashing the mint leaves with the back of a spoon to squeeze out all the juices. Add the rum, 125ml lime juice, lime zest and stir. Add enough soda water to bring the total volume up to 600ml. Taste the mixture – it should be sharp and sweet. Add the remaining 25ml lime juice if needed.
Pour the mixture into lolly moulds and freeze for 1 hr until the mixture is solid enough to hold the lolly sticks upright. Push the sticks 3/4 of the way into the lollies and return to the freezer for 6-8 hrs to freeze completely. To unmould, dip the outside of the lolly mould into hot water for 10 secs then use the sticks to pull each lolly out. The lollies can be kept wrapped individually in baking parchment in airtight bags in the freezer for up to 1 week.