In a blender, combine the watermelon, sugar and rum and blend them to homogeneity. Strain the mixture through a strainer, let it drain on its own, without stirring or "helping" it.
In tall glasses, put mint leaves on the bottom and rub them with a bit of brown sugar. Add slices of lime and crushed ice and fill the cups with the mixture with rum.
Stir and serve with a slice of lime.