Molecular hydrogen abundance in the dust-free damped Ly-alpha galaxy at z = 3.4

New results from the search for H2 absorption in the damped Ly-alpha galaxy at redshift z = 3.4 toward QSO 0000-2620 (z_em = 4.1) are reported. The high-resolution (lambda/Delta lambda = 48,000) spectra of Q0000-2620 were obtained using the Ultraviolet - Visual Echelle Spectrograph (UVES) on the 8.2m ESO Kueyen telescope. The ortho-H2 column density is found to be N(J=1) = (5.55 +/- 1.35) 10^{13} cm^{-2} (2sigma C.L.). The combination of N(J=1) with the limits available for other low rotational levels restricts the excitation temperature T_ex in the range (290-540) K. This gives the total H2 column density of N(H2) = (8.75 +/- 1.25) 10^{13} cm^{-2} and the corresponding fraction of hydrogen atoms bound in molecules of f(H2) = (6.8 +/- 2.0) 10^{-8}.