  Tunneling measurements of the temperature dependence of the electronic
density of states (DOS) of ultrathin bilayers of Pb and Ag reveal that their
superconducting energy gap, $\Delta (T)$, evolves similarly to BCS predictions
despite the presence of a large anomalous DOS at the Fermi energy that persists
as $T\to 0$. The gap ratio, ${2\Delta (0)}\over {k_BT_c}$, systematically
decreases more than 20% below the BCS universal value of 3.52 in bilayers with
the lowest superconducting transition temperatures, $T_c$; a behavior we deem
super weakly coupled. A semi-quantitative model suggests that the reduced gap
ratios result from a systematic depletion of the DOS available for pairing that
occurs with the growth of the anomalous DOS at the Fermi energy.
