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We present the detection analysis of a diffuse molecular cloud atz$_{abs}$=2.4636 towards the quasar SDSS J1513+0352(z$_{em}\,\simeq$ 2.68)observed with the X-shooter spectrograph(VLT). We measure very high columndensities of atomic and molecular hydrogen, with logN(HI,H$_2$)$\simeq$21.8,21.3. This is the highest H$_2$ column density evermeasured in an intervening damped Lyman-alpha system but we do not detect CO,implying log N(CO)/N(H$_2$) < -7.8, which could be due to a low metallicity ofthe cloud. From the metal absorption lines, we derive the metallicity to be Z$\simeq$ 0.15 Z$_{\odot}$ and determine the amount of dust by measuring theinduced extinction of the background quasar light, A$_V$ $\simeq$ 0.4. We alsodetect Ly-$\alpha$ emission at the same redshift, with a centroid located at amost probable impact parameter of only $\rho\,\simeq$ 1.4 kpc. We argue thatthe line of sight is therefore likely passing through the ISM of a galaxy asopposed to the CGM. The relation between the surface density of gas and that ofstar formation seems to follow the global empirical relation derived in thenearby Universe although our constraints on the star formation rate and on thegalaxy extent remain too loose to be conclusive. We study the transition fromatomic to molecular hydrogen using a theoretical description based on themicrophysics of molecular hydrogen. We use the derived chemical properties ofthe cloud and physical conditions (T$_k\,\simeq$90 K and n$\simeq$250 cm$^{-3}$derived through the excitation of H$_2$ rotational levels and neutral carbonfine structure transitions to constrain the fundamental parameters that governthis transition. By comparing the theoretical and observed HI column densities,we are able to bring an independent constraint on the incident UV flux, whichwe find to be in agreement with that estimated from the observed star formationrate.

Molecular gas and star formation in an absorption-selected galaxy: Hitting the bull's eye at z = 2.46