We report Monte Carlo simulations of the melting of a single-polymer crystallite. We find that, unlike most atomic and molecular crystals, such crystallites can be heated appreciably above their melting temperature before they transform to the disordered "coil" state. The surface of the superheated crystallite is found to be disordered. The thickness of the disordered layer increases with super-heating. However, the order-disorder transition is not gradual but sudden. Free-energy calculations reveal the presence of a large free-energy barrier to melting.