  We calculate the lowest-order contribution to the cross section for
simultaneous excitation of projectile and target nuclei in relativistic heavy
ion collisions. This process is, to leading order, non-classical and adds
incoherently to the well-studied semi-classical Weizs\"acker-Williams cross
section. While the leading contribution to the cross section is down by only
$1/Z_P$ from the semiclassical process, and consequently of potential
importance for understanding data from light projectiles, we find that phase
space considerations render the cross section utterly negligible.
