Scrub the limes vigorously in a large basin of warm water dotted with dishwashing liquid. Use an abrasive scrubber to clean the rind until it no longer feels waxy to the touch. Rinse the limes thoroughly; dry them. (Wash them carefully in this manner even if you're using organic or homegrown fruit.)

Put 2 cups sugar in the food processor. Using a sharp vegetable peeler or zester, remove the zest from 6 of the limes, letting the zest fall directly into the food processor. Be careful not to remove any of the bitter white pith.

Halve, juice, and strain enough limes to obtain 1 cup juice; set the juice aside.

Run the food processor until the zest is finely ground and the sugar is liquidy, 3 to 4 minutes. Old machines with worn blades take a bit longer and require several pauses to scrape down the bowl. Add the lime juice and process to blend.

Scrape the sugar-lime mixture into a large, non-aluminum bowl. Add the half-and-half and the salt, and stir to combine. Set the mixture aside for 15 to 30 minutes, stirring occasionally. It will thicken slightly.

Scrape the ice cream mixture through a fine-mesh sieve and discard the peel. At this point, the mixture can be sealed and refrigerated for a day before freezing. Don't worry if it separates; just stir to recombine.

Just before freezing, add the rum and stir to blend. Taste the mixture and adjust, if needed, with a dash more sugar or lime juice. The flavor should be round, and just a touch too sweet; the mixture should taste too sweet at room temperature if it is to taste perfect when frozen.

Freeze in an ice cream maker according to the manufacturer's instructions. Store with plastic wrap pressed directly on the surface. Let soften slightly before serving.