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.