We report on strong H2 and CO absorption from gas within the host galaxy of gamma-ray burst (GRB) 080607. Analysis of our Keck/LRIS afterglow spectrum reveals a very large HI column density (NHI = 10^22.70 cm^-2) and strong metal-line absorption at z_GRB = 3.0363 with a roughly solar metallicity. We detect a series of A-X bandheads from CO and estimate N(CO) = 10^16.5 cm^-2 and T_ex^CO > 100K. We argue that the high excitation temperature results from UV pumping of the CO gas by the GRB afterglow. Similarly, we observe H2 absorption via the Lyman-Werner bands and estimate N(H2) = 10^21.2 cm^-2 with T_ex^H2 = 10--300K. The afterglow photometry suggests an extinction law with R_V=4 and A_V=3.2 mag and requires the presence of a modest 2175A bump. Additionally, modeling of the Swift/XRT X-ray spectrum confirms a large column density with N(H) = 10^22.58 cm^-2. Remarkably, this molecular gas has extinction properties, metallicity, and a CO/H2 ratio comparable to those of translucent molecular clouds of the Milky Way, suggesting that star formation at high z proceeds in similar environments as today. However, the integrated dust-to-metals ratio is sub-Galactic, suggesting the dust is primarily associated with the molecular phase while the atomic gas has a much lower dust-to-gas ratio. Sightlines like GRB 080607 serve as powerful probes of nucleosynthesis and star-forming regions in the young universe and contribute to the population of dark GRB afterglows.
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