We study the dust depletion pattern in eight well separated components of the z=1.973, logN(HI)=20.83, damped Lyman-alpha system toward Q0013-004, four of which have detectable H2 absorption. The apparent correlation between the abundance ratios [Fe/S] and [Si/S] in the components indicates that the abundance pattern is indeed due to dust-depletion. In particular, we find evidence for depletion similar to what is observed in cold gas of the Galactic disk ([Fe/Zn]=-1.59, Fe/S=-1.74, Zn/S=-0.15, [Si/S]=-0.85) in one of the weakest components in which molecular hydrogen is detected with logN(H2)=16.5. This is the first time such a large depletion is seen in a DLA system. This observation supports the possibility that current samples of DLA systems might be biased against the presence of cold and dusty gas along the line of sight. The overall metallicities of this peculiar DLA system in which OI and CII are spread over 1050 km/s are [P/H]=-0.64, [Zn/H]=-0.74 and [S/H]=-0.82 relative to solar. The sub-DLA system at z=1.96753 has [P/H]>0.06, [Zn/H]>-0.02 and [S/H]>-0.18. The overall molecular fraction is in the range -2.7<logf<-0.6. CO is not detected (logN(CO)/N(HI)<-8) and HD could be present at z=1.97380. We show that the presence of H2 is closely related to the physical conditions of the gas: high particle density together with low temperature. The observed excitation of high J H2 levels and the molecular fraction show large variations from one component to the other suggesting that the UV radiation field is highly inhomogeneous throughout the system. Gas pressure, estimated from CI absorptions, is larger than what is observed in the ISM of our Galaxy. All this is probably a consequence of intense star-formation activity in the vicinity of the absorbing gas. (Abridged)
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