  We present transport and tunneling measurements of Pb-Ag bilayers with
thicknesses, $d_{Pb}$ and $d_{Ag}$, that are much less than the superconducting
coherence length. The transition temperature, $T_c$, and energy gap, $\Delta$,
in the tunneling Density of States (DOS) decrease exponentially with $d_{Ag}$
at fixed $d_{Pb}$. Simultaneously, a DOS that increases linearly from the Fermi
energy grows and fills nearly 40% of the gap as $T_c$ is 1/10 of $T_c$ of bulk
Pb. This behavior suggests that a growing fraction of quasiparticles decouple
from the superconductor as $T_c$ goes to 0. The linear dependence is consistent
with the quasiparticles becoming trapped on integrable trajectories in the
metal layer.
