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 introduces states within the gap. The integrated subgap DOS approaches 40% of the normal state value in the lowest T(c) film investigated (T(c) approximately 0.1 T(Pb)(c,bulk)). This behavior suggests that a growing fraction of quasiparticles decouple from the superconductor as T(c)-->0. The linear dependence is consistent with the quasiparticles becoming trapped on integrable trajectories in the metal layer.