We present 10 new gamma-ray burst (GRB) redshifts and another five redshift limits based on host galaxy spectroscopy obtained as part of a large program conducted at the Very Large Telescope (VLT). The redshifts span the range 0.345 < z < 2.54. Three of our measurements revise incorrect values from the literature. The homogeneous host sample researched here consists of 69 hosts that originally had a redshift completeness of 55% (with 38 out of 69 hosts having redshifts considered secure). Our project, including VLT/X-shooter observations reported elsewhere, increases this fraction to 77% (53/69), making the survey the most comprehensive in terms of redshift completeness of any sample to the full Swift depth, analyzed 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 <z> > 2.2, with the 10 new redshifts reducing it significantly. 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 that usually ascribed to lower metallicity.
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