Molecules at early epochs. VI. A search for molecular hydrogen in the Z = 3.391 damped Lyman alpha system toward Q 0000-263.

We report MMT observations at 1 A spectral resolution of the QSO 0000-263 (Z_em_ = 4.11) in the Ly-α forest between 4050 A and 4950 A. A computer code has been developed for setting the continuum level in the forest and for fitting blends with a minimum number of components. We have identified 221 absorption lines in the observed window and present evidence for high column-density systems at Z_abs_ = 3.3905 and 3.0533, the latter previously unreported. The Z = 3.39 system is the highest redshift damped Ly-α system known, having N(H) = 2 x 10^21^ cm^-2^. We find no convincing evidence for strong molecular hydrogen absorption in this system. At an adopted excitation temperature of T_ex_ = 1000K and a Doppler width of 25 km s^-1^, we set an upper limit to the fractional H_2_ abundance of f <= 3 X 10^-6^ which is lower than a typical Galactic value for high column-density interstellar clouds by a factor of nearly 10^5^.