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 $M_{W}=80.410\pm0.180 \ \mbox{GeV/c}^{2}$.