The phenomenology of an additional U(1) neutral gauge boson Z′ coupled to the third family of fermions is discussed. One might expect such a particle to contribute to processes where taus, b and t quarks are produced. Precision data from CERN LEP1 put severe constraints on the mixing and heavy-boson mass. We find that the effects of such a particle could not be observed at hadronic colliders, be it at the Fermilab Tevatron or the CERN LHC, because of the QCD background. At LEP2 and future e⁺e^- linear colliders, one could instead hope to observe such effects, in particular, for bb̄ final states.