  We calculate purely weak virtual one-loop corrections to the production cross
section of top-antitop pairs at the Large Hadron Collider via the gluon-gluon
fusion subprocess. We find very small negative corrections to the total cross
section, of order -0.6%, but significantly larger effects to the differential
one, particularly in the transverse momentum distribution, of order -5% to -10%
(in observable regions). In case of parity-conserving spin-asymmetries of the
final state, $\alpha_{\mathrm{S}}^2\alpha_{\mathrm{W}}$ corrections are
typically of a few negative percent, with the exception of positive and
negative peaks at $+12%$ and -5%, respectively (near where the tree-level
predictions change sign), while those arising in parity-violating asymmetries
(which are identically zero in QCD) are typically at a level of a few permille.
