Using the Green's function of the 3D heat equation, we develop an analytical account of the thermal behaviour of superconducting films subjected to electrical currents larger than their critical current in the absence of an applied magnetic field. Our model assumes homogeneity of films and current density, and besides thermal coefficients employs parameters obtained by fitting to experimental electrical field - current density characteristics at constant bath temperature. We derive both a tractable dynamic equation for the real temperature of the film up to the supercritical current density J^\ast (the lowest current density inducing transition to the normal state), and a thermal stability criterion that allows prediction of J^\ast . For two typical YBCO films, J^\ast predictions agree with observations to within 5%. These findings strongly support the hypothesis that a current-induced thermal instability is generally the origin of the breakdown of superconductivity under high electrical current densities, at least at temperatures not too far from Tc.