We present the redshift results from a Very Large Telescope (VLT) program aimed at optimizing the legacy value of the Swift mission: to characterize a homogeneous, X-ray selected, sample of 69 GRB host galaxies. Fifteen new redshifts have been secured, resulting in a 77% (53/69) 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 14% (5%) for z > 6 (z > 7). The mean redshift of the host sample is assessed to be . Using this more complete sample, we confirm previous findings that the GRB rate at high redshift () 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.