After the cooling period, pour the milk and sugar mixture into an ice cream maker (I have this one and it's great!). Churn until thickened, about 15-20 minutes.
To cut back on the amount of dairy in this recipe, I used almond milk. You can substitute with the type of milk you prefer. Whole milk will certainly make it creamier. As well, you can decrease the rum if you want it more subtle or try different varieties, like spiced or coconut.