We present the detection and detailed analysis of a diffuse molecular cloud at z abs = 2.4636 towards the quasar SDSS J 1513+0352 (z em 2.68) observed with the X-shooter spectrograph at the Very Large Telescope. We measure very high column densities of atomic and molecular hydrogen, with log N(H i, H 2) 21.8, 21.3. This is the highest H 2 column density ever measured in an intervening damped Lyman-α system but we do not detect CO, implying log N(CO)/N(H 2) < −7.8, which could be due to a low metallicity of the cloud. From the metal absorption lines, we derive the metallicity to be Z 0.15Z and determine the amount of dust by measuring the induced extinction of the background quasar light, A V 0.4. We simultaneously detect Lyα emission at the same redshift, with a centroid located at a most probable impact parameter of only ρ 1.4 kpc. We argue that the line of sight is therefore likely passing through the interstellar medium of a galaxy, as opposed to the circumgalactic medium. The relation between the surface density of gas and that of star formation seems to follow the global empirical relation derived in the nearby Universe although our constraints on the star formation rate and on the galaxy extent remain too loose to be conclusive. We study the transition from atomic to molecular hydrogen using a theoretical description based on the microphysics of molecular hydrogen. We use the derived chemical properties of the cloud and physical conditions (T k 90 K and n 250 cm −3) derived through the excitation of H 2 rotational levels and neutral carbon fine structure transitions to constrain the fundamental parameters that govern this transition. By comparing the theoretical and observed H icolumn densities, we are able to bring an independent constraint on the incident UV flux, which we find to be in agreement with that estimated from the observed star formation rate.