We report the detection of strong molecular hydrogen (H₂) absorption lines in the z = 2.34 damped Lyα absorber (DLA) toward QSO 1232+0815. This is the second detection in our survey for high-redshift molecules. The new ultraviolet spectrum of the QSO 1232+0815 at 0.9 Å resolution, taken with the Multiple Mirror Telescope Blue Channel Spectrograph, shows the υ = 0-0 up to 10-0 Lyman bands and also the Werner υ = 0-0 band of H₂ associated with the z = 2.34 DLA. We have estimated the total H₂ column density in this system. It ranges from ≈3 × 10¹⁹ cm^-2 to ≈3 × 10¹⁷ cm^-2 depending on the Doppler parameters, b = 6 or 10 km s^-1. Based on the best fit with b = 6 km s^-1, the estimated kinetic temperature is T_K ≈ 80 K. The measurements of the abundance of zinc and iron in the same DLA show that the metallicity measured by the relatively undepleted element zinc is [Zn/H] = -0.86, while the relative abundance ratio [Fe/Zn] is -1.04, indicating dust depletion. Combining our work with previous results on H₂ and relative heavy-element depletion, we find that there is a correlation between them, which suggests that the formation of H₂ on dust grains is perhaps the dominant formation process in high-redshift DLAs. Detections of strong absorption from the ground and excited states of neutral carbon (C I) in the z = 2.34 DLA are also presented. The excitation temperature between the two fine-structure levels of C I is 15.7 ± 3.5 K, an upper limit for the cosmic microwave background radiation temperature at z = 2.34. This value is consistent with the prediction by the standard big bang cosmology.