We present a new ultra-violet spectrum of the QSO 0013-004 with 0.9 AA resolution obtained with the MMT Blue spectrograph. The upsilon = 0 - 0, 1 - 0, 2 - 0 and 3 - 0 Lyman bands of H_2 associated with the z = 1.9731 damped Ly alpah absorption line system have been detected. The H_2 column density is N(H_2) = 6.9 (pm 1.6)times 10^{19} cm^{-2}, and the Doppler parameter b = 15pm 2 km/s. The populations of different rotational levels are measured and used to derive the excitation temperatures. The estimated kinetic temperature T_Ksim 70 K, and the total particle number density n(H) sim 300 cm^{-3}. The UV photoabsorption rate $beta_0 sim 6.7times 10^{-9}$ s^{-1}, about a factor of few times greater than that in a typical diffuse Milky Way interstellar cloud. The total hydrogen column density is $N(H) = 6.4(pm 0.5)times 10^{20} cm^{-2}$. The fractional H_2 abundance f = 2N(H_2)/(2N(H_2) + N(H I)) sim 0.22 pm 0.05 is the highest among all observed damped Lyal absorbers. The high fractional H_2 abundance is consistent with the inferred presence of dust and strong C I absorption in this absorber.
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