Combine the lime juice and the sugar in a small sauce pan and heat over medium heat until the sugar has dissolved; 1-2 minutes. Remove from heat, pour into a small bowl and refrigerate until completely cool.

Combine the blueberries and the sugar in a small saucepan. Warm over medium heat, stirring frequently, until the mixture is simmering and slightly reduced and the berries release their juices. Remove the pan from heat and set aside.

In a small bowl combine 2 Tablespoons of the milk with the cornstarch, whisk and set aside.

Combine the remaining milk and cream in a medium saucepan and place over medium heat. When the mixture is hot and begins to bubble, remove from the heat, add the mint and steep at room temperature for 20 minutes. Pour the mint infused milk into a bowl through a fine meshed sieve, pressing on the back of the mint with a wooden spoon to extract as much flavor as possible.

Fill a large bowl with ice water.

Return the mint infused milk to a medium saucepan and bring it to a low boil over medium heat. Remove from the heat and gradually whisk in the cornstarch mixture.

Return to a boil and cook over moderately high heat until the mixture is slightly thickened; about 30 seconds.Remove the the heat and pour into a medium bowl.

Set the bowl into the ice water bath to cool for about 20 minutes, whisking occasionally.

Combine the blueberry mixture with the cooled cream base in a blender and blend until the berries are fully incorporated.

Cover and refrigerate until well chilled, at least 4 hours or overnight.

Whisk the chilled ice cream base and add the lime syrup and rum. Pour the mixture into an ice cream maker and churn according to manufacturer's instructions. Transfer to a freezer safe container and freeze until firm, at least 4 hours.