We used HST STIS to obtain the spectrum of molecular hydrogen associated with the damped Lyα system at z_abs = 1.7765 toward the quasar Q1331+170 at z_em = 2.084. Strong H₂ absorption was detected, with a total H₂ column density of N(H₂) = (4.45 ± 0.36) × 10¹⁹ cm^-2. The molecular hydrogen fraction is f = 2N/(N + 2N) = 5.6% ± 0.7%, which is the greatest value reported so far in any redshifted damped Lyα 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 photoabsorption rate to be R_abs = (7.6 ± 2.4) × 10^-13 s^-1, corresponding to a local UV radiation field of J(1000 Å) ≈ 2.1 × 10^-3J_{1000 Å,} , where J_{1000 Å,} is the UV intensity at 1000 Å in the solar neighborhood. This is comparable to the metagalactic UV background intensity at this redshift and implies an extremely low star formation rate in the absorber's environment. We construct a simple model to describe the structure of the H₂ absorber, with a best-fit total hydrogen number density of n(H) ≈ 0.2 cm^-3 and an electron temperature of T_e ≈ 140 K. Assuming spherical symmetry, the mass of the H₂ cloud is estimated to be ≈6.5 × 10⁷ M, larger than the masses of most giant molecular clouds in the Milky Way and nearby galaxies. The extinction of Q1331+170 due to the intervening DLA is E_B-V ≈ 0.037, and we also find that the extinction by DLAs with firm H₂ detections is significantly greater than those for which only upper limits of f have been determined. The observed CO-to-H₂ column density ratio is N_CO/N < 2.5 × 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, we estimate the cosmic microwave background temperature to be T_CMB = (7.2 ± 0.8) K at z = 1.77654, consistent with the predicted value of 7.566 K from the standard cosmology.