  We report optical, near-infrared and X-ray observations of the afterglow of
GRB 050814, which was seen to exhibit very red optical colours. By modelling
its spectral energy distribution we find that z = 5.3 +/- 0.3. We next present
a carefully selected sample of 19 Swift GRBs, intended to estimate in an
unbiased way the GRB redshift distribution, including the mean redshift
(z_mean) as well as constraints on the fraction of high-redshift bursts. We
find that z_mean = 2.7 and that at least 5% of the GRBs originate at z > 5. The
redshift distribution of the sample is qualitatively consistent with models
where the GRB rate is proportional to the star formation rate in the Universe.
The high mean redshift of this GRB sample and the wide redshift range clearly
demonstrates the suitability of GRBs as efficient probes of galaxies and the
intergalactic medium over a significant fraction of the history of the
Universe.
