We report on strong H$_2$ 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 H I column density (N_H I = 10̂22.70 ± 0.15 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$^16.5±0.3$ cm$^-2$ and T $^CO$ $_ex$ > 100 K. We argue that the high excitation temperature results from UV pumping of the CO gas by the GRB afterglow. Similarly, we observe H$_2$ absorption via the Lyman-Werner bands and estimate N_H_2= 10^1̂.2 ± 0.2 cm^̂ with T_ex^H_=̂ 10-300 K. The afterglow photometry suggests an extinction law with R$_V$ ≈ 4 and A$_V$ ≈ 3.2 mag and requires the presence of a modest 2175 ̊A bump. Additionally, modeling of the Swift XRT X-ray spectrum confirms a large column density with N $_H$ = 10$^22.58±0.04$ cm$^-2$. Remarkably, this molecular gas has extinction properties, metallicity, and a CO/H$_2$ 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.