This is the second paper of a series reporting on the results from a survey conducted with the ESO VLT/X-shooter spectrograph. We target high-metallicity damped Lyman α absorbers (DLAs) with the aim of investigating the relation between galaxies detected in emission and those detected in absorption. Here, we report on the discovery of the galaxy counterpart of the zabs= 2.58 DLA on the line-of-sight to the z= 3.07 quasar SDSS J 091826.16+163609.0 (hereafter Q 0918+1636). The galaxy counterpart of the DLA is detected in the [O iii]λ5007 and [O ii]λλ3726, 3729 emission lines redshifted into the NIR at an impact parameter of 2.0 arcsec (16 kpc at z= 2.58). Lyα emission is not detected down to a 3σ detection limit of 5 × 10−18 erg s−1 cm−2, which, compared to the strength of the oxygen lines, implies that Lyα emission from this galaxy is suppressed by more than an order of magnitude. The DLA has one of the highest metallicities measured so far at comparable redshifts. We find evidence for substantial depletion of refractory elements on to dust grains. Fitting the main metal line component of the DLA, which is located at zabs= 2.5832, we measure the metal abundances from Zn ii, S ii, Si ii, Cr ii, Mn ii, Fe ii and Ni ii to be −0.12 ± 0.05, −0.26 ± 0.05, −0.46 ± 0.05, −0.88 ± 0.05, −0.92 ± 0.05, −1.03 ± 0.05 and −0.78 ± 0.05, respectively. In addition, we detect absorption in the Lyman and Werner bands of molecular hydrogen (H2), which represents the first detection of H2 molecules with X-shooter. The background quasar Q 0918+1636 is amongst the reddest QSOs at redshifts 3.02 < z < 3.12 from the SDSS catalogue. Its UV to NIR spectrum is well fitted by a composite QSO spectrum reddened by SMC-/LMC-like extinction curves at zabs= 2.58 with a significant amount of extinction given by AV≈ 0.2 mag. This supports previous claims that there may be more metal-rich DLAs missing from current samples due to dust reddening of the background QSOs. The fact that there is evidence for dust both in the central emitting regions of the galaxy (as evidenced by the lack of Lyα emission) and at an impact parameter of 16 kpc (as probed by the DLA) suggests that dust is widespread in this system.