  The production of top quark pairs accompanied by a hard gluon in $e^+e^-$
annihilation is studied including next-to-leading order corrections in the
strong coupling. At leading order, the fraction r of $t\bar{t}g$ events with
respect to all $t\bar{t}$ events is computed analytically as a function of the
minimal gluon energy. Next-to-leading order results for r are given for
center-of-mass energies of 0.5 and 1 TeV. We further calculate the differential
distribution of r with respect to several variables, including the top quark
energy and the $t\bar{t}$ invariant mass. We then investigate how our results
depend on the choice of the renormalization scheme for the top quark mass by
comparing results expressed in terms of either the pole mass or the $\msbar$
mass. Finally we estimate the sensitivity of the fraction r on the value of the
running top quark mass at a scale of 1 TeV.
