  Thin films of $\alpha$-MoGe show progressively reduced $T_{c}$'s as the
thickness is decreased below 30 nm and the sheet resistance exceeds 100
$\Omega/\Box$. We have performed far-infrared transmission and reflection
measurements for a set of $\alpha$-MoGe films to characterize this weakened
superconducting state. Our results show the presence of an energy gap with
ratio $2\Delta_0/k_BT_{c} = 3.8 \pm 0.1$ in all films studied, slightly higher
than the BCS value, even though the transition temperatures decrease
significantly as film thickness is reduced. The material properties follow
BCS-Eliashberg theory with a large residual scattering rate except that the
coherence peak seen in the optical scattering rate is found to be strongly
smeared out in the thinner superconducting samples. A peak in the optical mass
renormalization at $2\Delta_0$ is predicted and observed for the first time.
