In this paper we simulate a special type of Stefan problem in large-scale superconducting magnet systems in which superfluid helium (He II) is used as the coolant for the system. Liquid helium in a narrow channel called a "cable-in-conduit" conductor (CICC) is used to remove the heat load from the conductors. Liquid helium exhibits a phase change transition to normal helium (He I) when its temperature rises above the lambda point (2.716 K under saturated vapor pressure). A simple one-dimensional model is described to analyze this special He II/He I Stefan problem. A moving mesh technique is used to solve this model to improve the numerical efficiency compared with front-tracking methods. The results illustrate the simplicity and efficiency of this model. (c) 2006 Elsevier Ltd. All rights reserved.