  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
$t\bar t\to W^+bW^-\bar b\to l\nu b q q^\prime \bar b$ independent of the
lepton flavor, and results in a large acceptance for $W\to\tau\nu$ 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.)$\pm 10.8$(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.
