  We consider the quantum treatment of the rolling tachyon background
describing the decay of D-branes in the limit of weak string coupling. We focus
on the propagation of an open string in the fluctuating background and show how
the boundary string action is modified by quantum effects. A bilocal term in
the boundary action is generated which, however, does not spoil the vanishing
of the $\beta$ function at one loop. The propagation of an open string for
large times is found to be very strongly suppressed.
