We search for a third generation vector leptoquark (VLQ3) that decays to a b quark and tau lepton usingthe CDF II detector and 320 pb-1 of integrated luminosity from the Fermilab Tevatron. Observing anumber of events in agreement with standard model expectations, we obtain, assuming Yang-Mills (minimal) couplings, the most stringent upper limit on the VLQ3 pair production cross section of 344 fb(493 fb) and lower limit on the VLQ3 mass of 317 GeV/c2 (251 GeV/c2) at 95% C.L.