We have used the Collider Detector at Fermilab (CDF-II) to search for the flavor-changing neutral-current (FCNC) top-quark decay t→Zc using a technique employing ratios of W and Z production, measured in pp data corresponding to an integrated luminosity of 1.52  fb−1. The analysis uses a comparison of two decay chains, pp→tt→WbWb→ℓνbjjb and pp→tt→ZcWb→ℓℓcjjb, to cancel systematic uncertainties in acceptance, efficiency, and luminosity. We validate the modeling of acceptance and efficiency for lepton identification over the multiyear data set using another ratio of W and Z production, in this case the observed ratio of inclusive production of W to Z bosons. To improve the discrimination against standard model backgrounds to top-quark decays, we calculate the top-quark mass for each event with two leptons and four jets assuming it is a tt event with one of the top quarks decaying to Zc. For additional background discrimination we require at least one jet to be identified as originating from a b quark. No significant signal is found and we set an upper limit on the FCNC branching ratio Br(t→Zc) using a likelihood constructed from the ℓℓcjjb top-quark mass distribution and the number of ℓνbjjb events. Limits are set as a function of the helicity of the Z boson produced in the FCNC decay. For 100% longitudinally-polarized Z bosons we find limits of 8.3% and 9.3% (95% C.L.) depending on the assumptions regarding the theoretical top-quark pair production cross section.