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 pp̅ data corresponding to an integrated luminosity of 1.52  fb[superscript -1]. The analysis uses a comparison of two decay chains, pp̅ →tt̅ →WbWb→ℓνbjjb and pp̅ →tt̅ →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 tt̅ 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.Academy of FinlandSlovak R&D AgencyPrograma Consolider-Ingenio 2010, SpainMinisterio de Ciencia e Innovación, SpainRussian Foundation for Basic ResearchInstitut National de Physique Nucleaire et Physique des Particules/CNRSRoyal Society, United KingdomScience and Technology Facilities Council, United KingdomKorean Research FoundationKorean Science and Engineering FoundationBundesministerium für Bildung und Forschung, GermanyAlfred P. Sloan FoundationSwiss National Science FoundationNational Science Council of the Republic of ChinaNatural Sciences and Engineering Research Council of CanadaMinistry of Education, Culture, Sports, Science and Technology of JapanItalian Istituto Nazionale di Fisica NucleareNational Science FoundationUnited States Department of Energy