We have detected in high-resolution spectra of the quasar Q0347-3819 obtained with the UV-Visual Echelle Spectrograph (UVES) at the VLT/Kueyen telescope over 80 absorption features in the Lyman and Werner H2 bands at the redshift of a damped Lyα (DLA) system at zabs = 3.025. At the same redshift, numerous absorption lines of atoms and low ions (H I, D I, C II, C II*, N I, O I, Si II, P II, Ar I, Cr II, Fe II, and Zn II) were identified. The zabs = 3.025 system spans over ~80 km s-1 and exhibits a multicomponent velocity structure in the metal lines. The main component at zabs = 3.024855 shows a total H2 column density N(H2) = (4.10 ± 0.21) × 1014 cm-2 and a fractional molecular abundance fH2 = (1.94 ± 0.10) × 10-6 derived from the H2 lines arising from the J = 0-5 rotational levels of the ground electronic vibrational state. In the second strong component at zabs = 3.024684 we were able to set a stringent upper limit to fH2 of 3.3 × 10-9 (3 σ c.l.). The population of the J levels can be represented by a single excitation temperature of Tex = 825 ± 110 K. The ortho-H2 : para-H2 ratio equals 3.1 ± 0.3. This ratio is comparable to that calculated from hot H2 formation upon grain surfaces (ortho-H2 : para-H2 = 3 : 1) but is higher than the "freezeout" quantity (ortho-H2 : para-H2 = 1 : 4) predicted for cosmological epochs at z < 20, implying recent formation of the hydrogen molecules following photodissociation of pristine H2. In the main component the properties of the neutral atoms and low ions can be described with the same broadening parameter b ≃ 3 km s-1 of the H2 cloud. The rate at which UV photons are absorbed in the Lyman and Werner bands is found to be β0 ≃ 2 × 10-9 s-1. Taking into account that similar β0 values are observed in the Galactic diffuse molecular clouds, it shows that the UV radiation fields in the DLA and in the Milky Way are very much alike. We also found the H2 formation rate of RnH ≃ 3 × 10-16 s-1 at zabs = 3.025, which is consistent with observations in the Galactic disk diffuse clouds where nH2 is low. Comparing the relative rotational population ratios of H2 with the measured C II*/C II ratio, we can infer that nH ≃ 6 cm-3 in the main component, which gives a size of D ≃ 14 pc along the line of sight. The mean value of the dust-to-gas ratio estimated for the whole zabs = 3.025 system from the [Cr/Zn] ratio is equal to = 0.032 ± 0.005 (in units of the mean Galactic interstellar medium value). For the first time we unambiguously reveal a pronounced [α-element/iron peak] enhancement of [O,Si/Zn] = 0.6 ± 0.1 (1 σ c.l.) at high redshift. The simultaneous analysis of metal and hydrogen lines leads to D/H = (3.75 ± 0.25) × 10-5. This value is consistent with standard big bang nucleosynthesis if the baryon-to-photon ratio, η, lies within the range 4.37 × 10-10 ≲ η ≲ 5.32 × 10-10, implying 0.016 ≲ Ωbh ≲ 0.020.