A regime of the plasma wave-field accelerator (PWFA) is proposed, in which a high-intensity electron beam is used to excite extremely nonlinear, transverse motion-dominated plasma oscillations. Through computational analysis of the plasma electron motion and the associated wake fields, it is shown that if the beam is dense enough to eject nearly all of the plasma electrons from the beam channel, then the short-range wake fields are of excellent quality for acceleration and focusing of electron beams. These results clear up many conceptual difficulties with the practical realization of a PWFA.