An analysis is presented of the thermal instability which occurs in a discharge in a molecular gas in which the vibrational temperature is very different from the translational temperature because of the strong temperature dependence of the rate constant for vibrational relaxation of the molecules. The conditions for the onset of this instability, which is manifested by an abrupt increase in the temperature drop in the discharge tube, are found for the case of a diffusion discharge, in which the volume neutralization of charged particles is inconsequential. It is suggested that these conditions be used to determine the rate constants for vibrational relaxation of molecules which are known only approximately.