In the limit of high vibrational and electronic principal quantum numbers, a semiclassical model for the auto-ionization of diatomic molecules has been constructed. The molecular vibrations are treated as classical oscillators, whose parameters are chosen from a knowledge (which can be taken either from theory or experiment) of the positions of the quantal vibrational levels and the position and depth of the minimum of the adiabatic potential curve of the residual molecular ion. The theory is designed for the case where vibrational quantum numbers are greater than about 10. Electronic quantum numbers are restricted by the requirement that they be high enough so that (i) the excited-electron-core-electron interaction can be represented by the monopole term in the region where the excited electron is always farther from either nucleus than the core electron is, and (ii) the excited-electron-nuclear interactions can be represented by the monopole terms in all regions. An advantage of the theory over the perturbed-stationary-state theory is that its validity extends into the region of very high electronic quantum numbers (n=100), where the electron and nuclear velocities are comparable and the Born-Oppenheimer theory is not valid. Numerical estimates for the auto-ionization rates are presented for several sample cases for vibrations in the neighborhood of n'=10, excited electrons from n=10 to 20, and zero-energy ejected electrons. Numerical results are also included for the lower vibrations (n'<=5), n=8, 9, 10 excited electrons, and zero-energy ejected electrons, and comparison is made with experimental and other theoretical results.