We report the serendipitous detection of an H-2-bearing damped Lya absorber at z = 0.576 in the spectrum of the QSO J0111-0316 in the Cosmic Ultraviolet Baryon Survey. Spectroscopic observations from Hubble Space Telescope-COS in the far-ultraviolet reveal a damped absorber with log[N(H I)/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 Delta nu 60 km s(-1), with >99.9% of the total H-2 column density concentrated in one component. At a metallicity of approximate to 50% of solar, there is evidence for Fe enhancement and dust depletion, with a dust-to-gas ratio kappa(O) approximate to 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 Delta nu(g) <= 300 km s(-1) from the absorber. The closest is a massive, early-type galaxy at d = 41 pkpc that contains approximate to 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 starforming gas.