This is the final paper in a series dealing with the spectrum of the beat signal of a ring-laser gyro. The remaining region of operation of the gyro for which an approximate expression for the spectrum has not been obtained, i.e., when the laser rotation rate is close to the edge of the unlocked region, is examined here using an approach of a different character to those developed in Papers I and II. The method is based on solving the original Langevin equation for the phase by a variation-of-constants method. The need to adopt a consistent interpretation of such equations is emphasized; the Stratonovich interpretation is used here. The final results are obtained through an iterative procedure which converges much more rapidly than the continued-fractions method of Paper II. The lower-order result yields an approximate analytic expression for the beat signal spectrum which explicitly exhibits the harmonic structure of the spectrum for laser rotation rates close to the edge of the unlocked region. Numerical results are also presented showing the convergence of the overall procedure, and the accuracy of the approximate analytic results, for noise values appropriate to the ring-laser gyro.