This paper presents 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 $W
\rightarrow \mu\nu$ events recorded in an integrated luminosity of 19.7
pb$^{-1}$ gives
$M_{W}^{\mu}=80.310\pm0.205\mbox{~(stat.)}\pm0.130\mbox{~(syst.)}  \
\mbox{GeV/c}^{2}$. A fit to the transverse mass spectrum of a sample of 5718 $W
\rightarrow e\nu$ events recorded in 18.2 pb$^{-1}$ gives
$M_{W}^{e}=80.490\pm0.145\mbox{~(stat.)}\pm0.175\mbox{~(syst.)}  \
\mbox{GeV/c}^{2}$. Combining the muon and electron  results, accounting for
correlated uncertainties, yields $M_{W}=80.410\pm0.180 \ \mbox{GeV/c}^{2}$.