  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)
