  The top quark cross section close to threshold in $e^+e^-$ annihilation is
computed including the summation of logarithms of the velocity at
next-to-next-to-leading-logarithmic order in QCD. The remaining theoretical
uncertainty in the normalization of the total cross section is at the few
percent level, an order of magnitude smaller than in previous
next-to-next-to-leading order calculations. This uncertainty is smaller than
the effects of a light standard model Higgs boson.
