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We aim at deriving the physical conditions in the neutral gas associated with damped Lyman-α systems using observation and analysis of H₂ and CI absorptions. We obtained a high-resolution VLT-UVES spectrum of the quasar Q2348-011 over a wavelength range that covers most of the prominent metal and molecular absorption lines from the logN(HI)=20.50±0.10 damped Lyman-α system at z_abs=2.4263. We detected H₂ in this system and measured column densities of H₂, CI, CI^*, CI^**, SiII, PII, SII, FeII, and NiII. From the column density ratios and, in particular, the relative populations of H₂ rotational and Ci fine-structure levels, we derived the physical conditions in the gas (relative abundances, dust-depletion, particle density, kinetic temperature, and ionising flux) and discuss physical conditions in the neutral phase. Molecular hydrogen was detected in seven components in the first four rotational levels (J=0-3) of the vibrational ground state. Absorption lines of H₂ J=4 (resp. J=5) rotational levels are detected in six (resp. two) of these components. This leads to a total molecular fraction of logf≃-1.69^+0.37_–0.58. Fourteen components are needed to reproduce the metal-line profiles. The overall metallicity is found to be -0.80, -0.62, -1.17±0.10 for, respectively, [Si/H], [S/H] and [Fe/H]. We confirm the earlier findings that there is a correlation between logN(FeII)/N(SII) and logN(SiII)/N(SII) from different components indicative of a dust-depletion pattern. Surprisingly, however, the depletion of metals onto dust in the H₂ components is not large in this system: [Fe/S]=-0.8 to -0.1. The gas in H₂-bearing components is found to be cold but still hotter than similar gas in our Galaxy (T>130K, instead of typically 80K) and dense (n∼100-200cm^–3). There is an anti-correlation (R=-0.97) between the logarithm of the photo-absorption rate, logβ₀, and logN(H₂)/N(CI) derived for each H₂ component. We show that this is mostly due to shielding effects and imply that the photo-absorption rate β₀ is a good indicator of the physical conditions in the gas. We find that the gas is immersed in an intense UV field, about one order of magnitude higher than in the solar vicinity. These results suggest that the gas in H₂-bearing DLAs is clumpy, and star-formation occurs in the associated object.