  We present a study of the early build-up of dust in high redshift galaxies.
The study is based on the analysis of 38 Damped Ly alpha systems (DLAs) for
which we derive the fraction of iron atoms in dust form, f_{Fe}. The sample is
representative of metal-poor galaxies in the redshift range 0.6 </= z </= 3.4
selected on the basis of their absorption HI column density (N(HI) >/= 2 x
10^{20} atoms cm^{-2}). We find that the dust fraction increases with
metallicity, from f_{Fe}~0 at [Fe/H] ~ -2 dex, up to f_{Fe} ~ 0.9 at solar
metallicity; the increase is fast below [Fe/H] ~ -1 dex and mild at higher
metallicities. We also find some evidence for an increase of f_{Fe} with cosmic
time; a large fraction of the systems younger than ~3 Gyr has f_{Fe} </~ 0.5.
These results indicate the dust-to-metal ratio increases in the course of
chemical evolution, at variance with the hypothesis of an approximately
constant dust-to-metal ratio, commonly adopted in models of galactic evolution.
This hypothesis is consistent with local and high-redshift data only when the
metallicity is relatively high ([Fe/H] >/~ -1 dex). The results of this work
suggest that the main mechanisms of dust formation may be rather sensitive to
the level of metallicity attained by a galaxy in the course of its chemical
evolution. A metallicity-dependent dust production by SNe II seems to be the
most promising mechanism for explaining the rise of f_{Fe} at [Fe/H] </~ -1
dex.
