We present an analysis of top-antitop quark production and decay into a tau lepton, tau neutrino, and bottom quark using data from 9  fb-1 of integrated luminosity at the Collider Detector at Fermilab. Dilepton events, where one lepton is an energetic electron or muon and the other a hadronically decaying tau lepton, originating from proton-antiproton collisions at s=1.96  TeV, are used. A top-antitop quark production cross section of 8.1±2.1  pb is measured, assuming standard-model top quark decays. By separately identifying for the first time the single-tau and the ditau components, we measure the branching fraction of the top quark into the tau lepton, tau neutrino, and bottom quark to be (9.6±2.8)%. The branching fraction of top quark decays into a charged Higgs boson and a bottom quark, which would imply violation of lepton universality, is limited to be less than 5.9% at a 95% confidence level [for B(H-→τν¯)=1].