Phase–change problems often involve discontinuities in the thermal properties at the phase–change boundary. This feature needs to be handled carefully when seeking a numerical solution based on a fixed space grid. Of particular concern are discontinuities in the thermal conductivity. In the context of a control–volume finite–difference solution, the requirement is an appropriate approximation of the conductivity values at the control–volume interfaces. In this article, using the Kirchhoff transformation, an approximation for the interface conductivity is developed. The approach is tested on a range of one– and two–dimensional, steady and transient phase–change problems. In addition, a discussion on the extension of the approach to finite–element schemes is included.