We demonstrate that a self-consistent circularly polarized wave in an otherwise field-free homogeneous cold plasma is unstable to small amplitude perturbations. For either an electron-positron plasma or an electorn-proton plasma the instability rate is at least about the order of the effective plasma frequency when the bulk flow speed is zero. For finite bulk flow speeds of the plasma, we show that the electron-positron plasma is unstable – again with a growth rate of the order of the effective plasma frequency; we also show that the electron-proton plasma is unstable (at least at small wave number, k) with a growth rate proportional to k. The instability rates we calculate are conservative, for other modes not investigated here may be more unstable. The results of these calculations bear directly on our understanding of plasma systems thought to be driven by large amplitude waves.