Add lime, sugar, and rum to a saucepan and bring to boil. Simmer for 15 minutes until the liquid reaches a syrupy consistency. Remove from heat and allow to cool.
Add cream and milk to a large saucepan on medium heat.
Meanwhile, whisk egg yolks and lime zest until yolks are pale.
As the milk starts to simmer, remove from heat and pour half a cup of the liquid into the yolks, all the while whisking.
Pour the yolks back into the milk and whisk until combined.
Continue stirring the milk on a medium heat until it thickens enough to coat the back of a spoon.
Remove from heat and stir in the lime syrup.
Add chopped mint then add to the fridge and allow to cool for at least 2 hours. If you leave it overnight it will allow the mint flavour to infuse better.
Strain the custard base then add to your ice cream machine and churn as per the instructions. Scoop into an airtight container and freeze for at least 2 hours until hardened.
