  We present a measurement of the top-quark width using $t\bar{t}$ events
produced in $p\bar{p}$ collisions at Fermilab's Tevatron collider and collected
by the CDF II detector. In the mode where the top quark decays to a $W$ boson
and a bottom quark, we select events in which one $W$ decays leptonically and
the other hadronically~(lepton + jets channel) . From a data sample
corresponding to 4.3~fb$^{-1}$ of integrated luminosity, we identify 756
candidate events. The top-quark mass and the mass of $W$ boson that decays
hadronically are reconstructed for each event and compared with templates of
different top-quark widths~($\Gamma_t$) and deviations from nominal jet energy
scale~($\Delta_{JES}$) to perform a simultaneous fit for both parameters, where
$\Delta_{JES}$ is used for the {\it in situ} calibration of the jet energy
scale. By applying a Feldman-Cousins approach, we establish an upper limit at
95$\%$ confidence level~(CL) of $\Gamma_t <$ 7.6 GeV and a two-sided 68$\%$ CL
interval of 0.3 GeV $< \Gamma_t <$ 4.4 GeV for a top-quark mass of 172.5
GeV$/c^2$, which are consistant with the standard model prediction. This is the
first direct measurement of $\Gamma_t$ to set a lower limit with 68$\%$ CL.
