We present a measurement of the mass of the
$W$ boson using data collected with
the CDF detector during the 1992-93 collider run at the Fermilab Tevatron.   A
fit to the transverse mass spectrum of a sample of 3268 {\mbox{$W \rightarrow
\mu\nu$}} events recorded in an integrated luminosity  of 19.7 pb$^{-1}$ gives a
mass $M_{W}^{\mu}=80.310\pm0.205\mbox{~(stat.)}\pm0.130\mbox{~(syst.)}  \
\mbox{GeV/c}^{2}$. A fit to the transverse mass spectrum of a sample of 5718
{\mbox{$W \rightarrow e\nu$}} events recorded in 18.2 pb$^{-1}$ gives a mass
$M_{W}^{e}=80.490\pm0.145\mbox{~(stat.)}\pm0.175\mbox{~(syst.)} \
\mbox{GeV/c}^{2}$.  Combining the electron and muon results, accounting for
correlated uncertainties, yields a mass $M_{W}=80.410\pm0.180  \
\mbox{GeV/c}^{2}$.