We prove the existence of generalized weak solutions of a new model for the process of in situ vitrification. The model describes the steady process of vitrification of soil, i.e. the melting of soil by means of electrical current and subsequent solidification as a rock. This is the core of a new technology developed to treat buried low-level radioactive or toxic waste. It is a free boundary problem consisting of a degenerate elliptic equation for the electric potential coupled with a stationary Stefan problem for melting the soil, which results from the Joule heating. The degeneracy reflects the smallness of the ratio of the electrical conductivity of the soil to that of the molten rock. The existence result is proved by considering a sequence of approximate nondegenerate problems, obtaining the necessary a priori estimates and passing to the limit. We also establish a sufficient condition for the solution to exhibit a molten region.