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

New results from the search for H-2 absorption in the damped Lyalpha galaxy at redshift z = 3.4 toward QSO 0000-2620 (z,, = 4.1) are reported. The high-resolution (lambda/Deltalambda = 48, 000) spectra of Q0000-2620 were obtained using the Ultraviolet - Visual Echelle Spectrograph (UVES) on the 8.2m ESO Kueyen telescope. The ortho-H-2 column density is found to be N(J = 1) = (5.55 +/- 1.35) X 10111 cm(-2) (2sigma C.L.). The combination of N(J = 1) with the Emits available for other low rotational levels restricts the excitation temperature T in the range (290 - 540) K. This gives the total H2 column density Of N(112) = (8.75 +/- 1.25) x 10(13) cm(-2) and the corresponding fraction of hydrogen atoms bound in molecules Of f (112) (6.8 +/- 2.0) x 10(-8).