The problem of the inward solidification of a spherical or cylindrical body of molten material, initially at its uniform fusion temperature, when the outside is suddenly cooled, is considered. A complete asymptotic theory is developed for the case when the parameter $\lambda$, which measures the ratio of latent heat to sensible heat of the substance, is large. Uniformly valid approximations to the solution are found everywhere, for all time t$^*$, up to the instant t$^*$ = t$^*_e$, at which the material is completely frozen. Though many of the results have been obtained previously, the treatment of the final freezing of the central core as t$^*\rightarrow$ t$^*_e$ is new. For the cylinder, the novel approach enables asymptotic solutions to be obtained, when t$^*\rightarrow$ t$^*_e$, for the first time.