We used HST/STIS to obtain the spectrum of molecular hydrogen associated with the damped Ly$alpha$ system at $z_{rm abs}=1.7765$ toward the quasar Q1331+170 at $z_{rm em}=2.084$. Strong ${rm H}_2$ absorption was detected, with a total ${rm H}_2$ column density of $N({rm H}_2)=(4.45pm 0.36)times 10^{19} {rm cm^{-2}}$.The molecular hydrogen fraction is $f_{{rm H}_2}=frac{2N_{rm H_2}}{N_{rm HI}+2N_{rm H_2}}=(5.6pm 0.7)%$, which is the greatest value reported so far in any redshifted damped Ly$alpha$ system. This results from the combined effect of a relatively high dust-to-gas ratio, a low gas temperature, and an extremely low ambient UV radiation field. Based on the observed population of $J$ states, we estimate the photo-absorption rate to be $R_{rm abs}=(7.6pm 2.4)times 10^{-13} {rm s^{-1}}$, corresponding to a local UV radiation field of $J(1000{rm AA})approx 2.1times 10^{-3} J_{1000{rm AA},odot}$, where $J_{1000{rm AA},odot}$ is the UV intensity at $1000 AA$ in the solar neighborhood. This is comparable with the metagalactic UV background intensity at this redshift, and implies an extremely low star formation rate in the absorbers environment. The observed CO-to-H$_2$ column density ratio is $frac{N_{rm CO}}{N_{rm H_2}}<2.5times 10^{-7}$, which is similar to the value measured for diffuse molecular clouds in the Galactic ISM. Finally, applying the inferred physical conditions to the observed C I fine structure excitation (Songaila {it et al.} 1994), we estimate the cosmic microwave background temperature to be $T_{rm CMB}=(7.2pm 0.8) {rm K}$ at $z=1.77654$, consistent with the predicted value of $7.566 {rm K}$ from the standard cosmology.
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