  The effects of the dust on the determination of elemental abundances in
damped Ly alpha (DLA) absorbers are investigated. Relations between the
observed abundances measured in the gas phase and the overall abundances (gas
plus dust) are derived as a function of dust-to-gas ratio, metallicity,
element-to-element abundance pattern, average extinction coefficient of dust
grains, and chemical composition of dust grains. A method is presented for
determining dust-to-gas ratios, dust-to-metals ratios, and dust-corrected
relative abundances in DLA absorbers by assuming dust of Galactic type and
constant abundance ratios between iron-peak elements. The method is applied to
a sample of 17 DLA absorbers with available Zn, Cr and/or Fe measurements. The
resulting dust-to-gas ratios are mostly distributed between 2% and 25% of the
Galactic value, in good quantitative agreement with the results from reddening
studies of QSOs with foreground DLA absorption. A correlation is found between
dust-to-gas ratio and metallicity in DLA galaxies, with a typical
dust-to-metals ratio of ~ 60% the Galactic value. The derived dust-to-metals
ratios are then used to correct from the effects of dust the abundance ratios
[Si/Fe], [S/Fe], [Ti/Fe], [Mn/Fe], [Ni/Fe] available for a sub-sample of 9
absorbers. The [alpha/Fe] ratios corrected from dust do not show the
enhancement characteristic of metal-poor Galactic stars, but instead have
essentially solar values, within +/- 0.2 dex. This suggests that the chemical
history of DLA absorbers is different from that experienced by the Milky Way.
Evidences that point to dwarf galaxies, rather than to spiral galaxies, as
important contributors to the DLA phenomenon are summarized.
