The stability of a finite amplitude circularly polarized Alfv¿n wave of wave number k0 is studied by using the two-fluid isentropic equations. Linear perturbation analysis, involving two sideband transverse waves having wave numbers k0¿k and a longitudinal wave with wave number k, is used to find the exact sixth-order dispersion relation. The analysis is then limited to the case where k≪k0. The resulting fourth-order dispersion relation is examined analytically and numerically, and a surface is found that separates stable and unstable regions in parameter space. This surface describes the boundary between stable and unstable regions not only for k≪k0 but for the entire branch of the dispersion relation which extends to k=0. We refer to this branch as the modulation branch and the corresponding instability as a modulation instability. A sufficient condition for modulation stability is found to be vϕ0cs for right-hand polarized waves, where vϕ0 and cs are the phase velocity of the unperturbed wave and the unperturbed sound speed, respectively. Modulation wave amplitudes and growth rates are given.