An assessment is presented for the thermal attack on the MARK-I boiling water reactor steel containment shell by core melt materials ejected from the vessel in a severe accident. The cooling of the core melt as it spreads and transfers heat to the concrete floor of the drywell is evaluated. It is found that the melt temperature may reach the freezing point before the melt contacts the shell, particularly if the melt was mostly oxidic or was ejected at moderate rates. The heat fluxes from the melt to the liner that can be withstood are evaluated, with and without a pool of water overlying the melt. With water above the melt, if the superheat in a mostly metallic melt is moderate to allow for the formation of a crust at the interface with the shell, the shell may survive the attack by a shallow melt layer (up to 10 cm deep). The potential for survival is much better if the melt was composed mostly of oxidic materials.