We present a measurement of the top-quark mass in $p\bar p$ collisions at \hbox{$\sqrt{s}=1.96$} TeV which uses events with an inclusive signature of missing transverse energy and jets. % The event selection is sensitive to $t\bar t\to W^+bW^-\bar b\to lν b q q^\prime \bar b$ independent of the lepton flavor, and results in a large acceptance for $W\toτν$ decays. % All-hadronic $t\bar t$ decays and events with identified electrons or muons are vetoed to provide a statistically independent sample with respect to all previous measurements. % The top-quark mass is inferred from the distribution of the scalar sum of all jet transverse energies and the missing transverse energy. Using 311 pb$^{-1}$ of integrated luminosity recorded by the Collider Detector at Fermilab, we measure a top-quark mass $m_t = 172.3 ^{+10.8}_{-9.6}$(stat.)$± 10.8$(syst.) GeV/$c^2$. While the uncertainty on $m_t$ is larger than that of other measurements, the result is statistically uncorrelated with those of other methods, and thus can help to reduce the overall $m_t$ uncertainty when combined with other existing measurements. ; Comment: 8 pages, 4 figures