  We have used the Collider Detector at Fermilab (CDF II) to search for the
flavor-changing neutral-current (FCNC) top quark decay ${t\goes Zc}$ using a
technique employing ratios of $W$ and $Z$ production, measured in \ppbar data
corresponding to an integrated luminosity of 1.52 $fb^{-1}$. The analysis uses
a comparison of two decay chains, ${p\bar{p}\goes t\bar{t}\goes
WbWb\goes\ell\nu bjjb}$ and ${p\bar{p}\goes t\bar{t}\goes ZcWb\goes\ell\ell
cjjb}$, to cancel systematic uncertainties in acceptance, efficiency, and
luminosity. We validate the modeling of acceptance and efficiency for lepton
identification over the multi-year dataset 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 $t\bar{t}$ 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
$Br({t\goes Zc})$ using a likelihood constructed from the ${\ell\ell cjjb}$ top
quark mass distribution and the number of ${\ell\nu bjjb}$ 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.
