  To match the precision of present and future measurements of Z-boson
production at hadron colliders, electroweak radiative corrections must be
included in the theory predictions. In this paper we consider their effect on
the transverse momentum ($p_T$) distribution of Z bosons, with emphasis on
large $p_T$. We evaluate, analytically and numerically, the full one-loop
corrections for the parton scattering reaction $q\bar q \to Z g$ and its
crossed variants. In addition we derive compact approximate expressions which
are valid in the high-energy region, where the weak corrections are strongly
enhanced by logarithms of $\hat s/M_W^2$. These expressions include quadratic
and single logarithms as well as those terms that are not logarithmically
enhanced. This approximation, which confirms and extends earlier results
obtained to next-to-leading logarithmic accuracy, permits to reproduce the
exact one-loop corrections with high precision. Numerical results are presented
for proton-proton and proton-antiproton collisions. The corrections are
negative and their size increases with $p_T$. For the Tevatron they amount up
to -7% at 300 GeV. For the LHC, where transverse momenta of 2 TeV or more can
be reached, corrections up to -40% are observed. We also include the dominant
two-loop effects of up to 8% in our final LHC predictions.
