  We present the redshift results from a Very Large Telescope program aimed at
optimizing the legacy value of the Swift mission: to characterize a
homogeneous, X-ray selected, sample of 69 GRB host galaxies. 19 new redshifts
have been secured, resulting in a 83% (57/69) redshift completion, making the
survey the most comprehensive in terms of redshift completeness of any sample
to the full Swift depth, available to date. We present the cumulative redshift
distribution and derive a conservative, yet small, associated uncertainty. We
constrain the fraction of Swift GRBs at high redshift to a maximum of 10% (5%)
for z > 6 (z > 7). The mean redshift of the host sample is assessed to be <z> >
2.2. Using this more complete sample, we confirm previous findings that the GRB
rate at high redshift (z > 3) appears to be in excess of predictions based on
assumptions that it should follow conventional determinations of the star
formation history of the universe, combined with an estimate of its likely
metallicity dependence. This suggests that either star formation at high
redshifts has been significantly underestimated, for example due to a dominant
contribution from faint, undetected galaxies, or that GRB production is
enhanced in the conditions of early star formation, beyond those usually
ascribed to lower metallicity.
