  We present the detection analysis of a diffuse molecular cloud at
z$_{abs}$=2.4636 towards the quasar SDSS J1513+0352(z$_{em}\,\simeq$ 2.68)
observed with the X-shooter spectrograph(VLT). We measure very high column
densities of atomic and molecular hydrogen, with log
N(HI,H$_2$)$\simeq$21.8,21.3. This is the highest H$_2$ column density ever
measured 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 of
the 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 the
induced extinction of the background quasar light, A$_V$ $\simeq$ 0.4. We also
detect Ly-$\alpha$ emission at the same redshift, with a centroid located at a
most probable impact parameter of only $\rho\,\simeq$ 1.4 kpc. We argue that
the line of sight is therefore likely passing through the ISM of a galaxy as
opposed to the CGM. 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\,\simeq$90 K and n$\simeq$250 cm$^{-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 HI column 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.
