  We report the serendipitous detection of an H$_{2}$-bearing damped
Lyman-$\alpha$ absorber at z = 0.576 in the spectrum of the QSO J0111-0316 in
the Cosmic Ultraviolet Baryon Survey. Spectroscopic observations from HST-COS
in the far-ultraviolet reveal a damped absorber with log[N(HI)/cm^-2] = 20.1
+/- 0.2 and log[N(H$_{2}$)/cm^-2] = 18.97 (-0.06, +0.05). The diffuse molecular
gas is found in two velocity components separated by dv = 60 km/s, with >99.9%
of the total H$_{2}$ column density concentrated in one component. At a
metallicity of $\approx$ 50% of solar, there is evidence for Fe enhancement and
dust depletion, with a dust-to-gas ratio $\kappa_{\text{O}} \approx$ 0.4. A
galaxy redshift survey conducted with IMACS and LDSS-3C on Magellan reveals an
overdensity of nine galaxies at projected distance d <= 600 proper kpc (pkpc)
and line-of-sight velocity offset dv$_{g}$ <= 300 km/s from the absorber. The
closest is a massive, early-type galaxy at d = 41 pkpc which contains $\approx$
70% of the total stellar mass identified at d <= 310 pkpc of the H$_{2}$
absorber. The close proximity of the H$_{2}$-bearing gas to the quiescent
galaxy and the Fe-enhanced chemical abundance pattern of the absorber suggest a
physical connection, in contrast to a picture in which DLAs are primarily
associated with gas-rich dwarfs. This case study illustrates that deep galaxy
redshift surveys are needed to gain insight into the diverse environments that
host dense and potentially star-forming gas.
