The stability of a finite amplitude plane monochromatic circularly polarized Alv¿n wave is studied by using an approximate two-fluid model obtained by performing an amplitude and dual time scale expansion in which temporal changes, observed when moving with the wave, are assumed slow. The lowest order dispersive effects and coupling terms to sound waves are of the same order. A large-amplitude Alfv¿n wave obeying an approximate dispersion relation is an exact solution of the resulting model equations. Small perturbations of this solution consisting of two sideband Alv¿n waves and a sound wave are then introduced. The resulting dispersion relation, which is s of fourth order, is examined analytically for small wave amplitudes of the parent wave. It reveals different regions of stability and instability in a plot of wave number k0 of the unperturbed waves versus &bgr;0, the ratio of sound speed to Alfv¿n speed. For long wave lengths, the left-handed polarized mode is found to be stable for &bgr;0>1 and the right-handed polarized mode for &bgr;01. Decay instability is predicted near &bgr;0=0 and &bgr;0=1. Formulas for growth rates and unstable wave number bands are given.