Where does the phrase just deserts come from?

Why do we say that someone has gotten their just deserts? Does this turn of phrase have anything to do with dessert (“a sweet food eaten at the end of a meal”) or desert (“a dry land with few plants and little rainfall”)? In fact, the phrase employs neither of these words. Instead, it uses a completely unrelated word that happens to be pronounced like the word for sweets and spelled like the one for a dry place: desert, meaning “reward or punishment deserved or earned by one’s qualities or acts.” This little-used noun is, as you might have guessed, related to the English verb deserve. It has nothing to do with arid, dry land, or with cookies and ice cream.