  We consider non-singular bouncing cosmologies, such as the new ekpyrotic
model, in which the universe undergoes a slow contraction phase with equation
of state $w \gg 1$, followed by a bounce that occurs at a finite scale factor
when quantum gravity corrections are still negligible. Such a non-singular
bounce requires a violation of the null energy condition in which $w$ falls
below -1 at some time before the bounce. In this paper, we show that a
component of the adiabatic curvature perturbations, though decaying and
negligible during the ekpyrotic phase, is exponentially amplified just before
$w$ approaches -1, enough to spoil the scale-invariant perturbation spectrum.
We discuss how the problem may be avoided, for example, in singular bounces.
