  We present simultaneous optical and near-infrared (NIR) spectroscopy of 19
Swift GRB host galaxies with VLT/X-shooter with the aim of measuring their
redshifts. Galaxies were selected from The Optically Unbiased GRB Host (TOUGH)
survey (15 of the 19 galaxies) or because they hosted GRBs without a bright
optical afterglow. Here, we provide emission-line redshifts for 13 of the
observed galaxies with brightnesses between F606W > 27 mag and R=22.9 mag
(median R=24.6 mag). The median redshift is z=2.1 for all, and z=2.3 for the
TOUGH hosts. Our new data significantly improve the redshift completeness of
the TOUGH survey, which now stands at 77% (53 out of 69 GRBs). They furthermore
provide accurate redshifts for nine prototype-dark GRBs (e.g., GRBs 071021 at
z=2.452 and 080207 at z=2.086), which are exemplary of GRBs where redshifts are
challenging to obtain via afterglow spectroscopy. This establishes X-shooter
spectroscopy as an efficient tool for redshift determination of faint,
star-forming, high-redshift galaxies such as GRB hosts. It is hence a further
step towards removing the bias in GRB samples that is caused by optically-dark
events, and provides the basis for a better understanding of the conditions in
which GRBs form. The distribution of column densities as measured from X-ray
data (N_{H,X}), for example, is closely related to the darkness of the
afterglow and skewed towards low N_{H, X} values in samples that are dominated
by bursts with bright optical afterglows.
