  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 =
15\pm 2 km/s. The populations of different rotational levels are measured and
used to derive the excitation temperatures. The estimated kinetic temperature
T_K\sim 70 K, and the total particle number density n(H) \sim 300 cm^{-3}. The
UV photoabsorption rate $\beta_0 \sim 6.7\times 10^{-9}$ 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 Ly\al absorbers. The high
fractional H_2 abundance is consistent with the inferred presence of dust and
strong C I absorption in this absorber.
