We analyze theoretically a method for measuring the temperature dependence of the imaginary part of the gap at the gap edge, ${\ensuremath{\Delta}}_{2}(T)={\ensuremath{\Delta}}_{2}({\ensuremath{\Delta}}_{0}(T), T)$, based upon the behavior of the critical dc Josephson current carried by a superconducting tunnel junction in the presence of a high-frequency electromagnetic field. It is shown that the form of this temperature dependence provides a low-frequency probe of the effective electron-phonon coupling.