It is shown that the velocity threshold of the right-hand-polarized ion-ion resonant instability decreases with decreasing gyrofrequency values of the beam ions, and also, the system can become unstable for very small beam concentrations. In order to illustrate the results, a plasma consisting of a proton background and a very tenuous O+ ion beam is considered. It is shown that for an ion beam of nO+/np=10-4, the threshold velocity is VO+~1.36VA (VA is the Alfv¿n velocity), and the maximum growth rate is of the order of 10-3&OHgr;p (&OHgr;p is the proton gyrofrequency). We apply these results to high-speed solar-wind-type plasmas consisting of a proton background, an alpha particle beam, and a beam of O+ ions. It is also shown that the presence of alpha particles reduces the velocity instability threshold of the O+-p resonant instability. ¿ American Geophysical Union 1996