Norris et al, [1983> have recently shown that thermal (~1 eV) electrons are accelerated along fields lines (up to tens of electron volts) when intense ion cyclotron waves (ICW's) are simultaneously present in the equatorial magnetosphere. In the present paper a mechanism that explains these observations is proposed. It is shown that, owing to the small but finite parallel electric field developed by ICW's in a multicomponent plasma (e.g., containing minor He+ ions), electrons having parallel velocities close to the Alfven velocity (at the equator) can be trapped in the potential troughs of this ICW. Then, taking into account the inhomogeneity along geomagnetic field lines, it is shown that the increase of the parallel phase velocity of the ICW as it leaves the equator implies a parallel acceleration of these trapped electrons. The maximum parallel energy that is reached (a few tens of electron volts) is in agreement with observations; it is obtained by matching the trapping force (proportional to E∥) to the detrapping force due to the inhomogeneity along field lines.