Some heat and mass transport processes in subsea permafrost are discussed. An exact solution is given for a coupled model in which both heat and salt are transported solely by diffusion. The model describes the development of a thawed layer beneath the seabed after an ocean transgression over deeply frozen ground and is a generalization of the well-known Stefan problem. A stability calculation indicates that such a diffusive regime should be unstable to convection of the interstitial water in the thawed layer after a critical thickness has been reached. The theory is compared with observations from two sites in the Beaufort Sea, Alaska. In western Elson Lagoon, near Barrow, the diffusive regime may be stable, but at Prudhoe Bay it is not, and interstitial water motion is important. At Prudhoe Bay the salt and heat transport regimes are convective and diffusive, respectively.