  It has recently been shown that a new neutral vector boson $Z'$, with a mass
of order 1~TeV and weak couplings to quarks only, could explain both the
anomalous values of $R_b$ and $R_c$ and the apparent excess of large $E_T$ jet
events measured by the CDF collaboration. We calculate the effects of $Z'$
exchange on the $t\bar{t}$ production cross section at Tevatron $p \bar p$
collider energies, including next-to-leading-order QCD corrections. We find a
significant enhancement of the cross section and study the resulting $t \bar t$
invariant mass distribution, which could provide a decisive test of the $Z'$
model.
