Heat transfer correlation for the steady-state natural convection in a meniscus-shaped cavity was established. It was aimed at providing more accurate data for the contact melting processes in both cylindrical and spherical geometries. A closed-form exact conduction solution was also presented for the cylindrical case. It was found that normalized circumferential distribution of Nusselt number was approximately Nu/Nu(o) similar to cos phi, which confirmed successfully the shape-preserving character of the top solid surface. A sample application revealed that the convection melting in a cylindrical enclosure was estimated somewhat higher than that analyzed in the previous study. Admitting that natural convection is of secondary importance during contact melting, the present heat transfer correlation, combined with the elegant lubrication theory available elsewhere, would provide a more accurate prediction without solving complicated governing equations case by case.