We consider a mathematical model that describes the combined processes of heat conduction and electrical conduction in a body which may undergo a phase change as a result of the heat generated by the current, the so-called Joule heating. The model consists of a system of nonlinear partial differential equations with quadratic growth in the gradient. We prove that the model possesses a weak solution. Joule heating is generated by the resistance of materials to electrical current and is present in any electrical conductor operating at normal temperatures. The heating and melting of such conductors are usually undesirable side effects, but in electrical heaters the heating is welcome. The melting of the conductor is useful in fuses and is the basis of the industrially important process of electrical welding. Mathematical and computational problems related to models dealing with melting of materials as a result of volume heat sources were considered by Atthey [1] and by Crowley and Ockendon [6], among others. In [1] the heat source was assumed to be known. That is, the current density was assumed to be given beforehand. In [6] it was assumed to depend only on the temperature. Problems dealing with the combined heat and current flows, but without allowing for melting of the material,