Heat the water, sugar and mint leaves in a saucepan over medium heat; stir until the sugar is dissolved. Bring the mixture to a boil, reduce heat and simmer for 3 to 5 minutes to extract the mint flavour. Set the mixture aside to cool and strain out the mint leaves.
Pour the cooled mint mixture, lime zest, lime juice, sparkling water and rum into a bowl and mix well. Pour into the container of an ice cream maker and freeze according to the manufacturer's instructions. Serve immediately for a softer texture or freeze in a sealed container for a harder sorbet. Let hard-frozen sorbet thaw a few minutes before serving.
If citrus-flavoured sparkling water is available, substitute this in instead.