1995 The American Physical Society This paper presents a measurement of the mass of the W boson using data collected with the CDF detector during the 1992-1993 collider run at the Fermilab Tevatron. A fit to the transverse mass spectrum of a sample of 3268 W events recorded in an integrated luminosity of 19.7 pb-1 gives MW=80.3100.205 (stat)0.130 (syst) GeV/c2. A fit to the transverse mass spectrum of a sample of 5718 We events recorded in 18.2 pb-1 gives MWe=80.4900.145 (stat)0.175 (syst) GeV/c2. Combining the muon and electron results, accounting for correlated uncertainties, yields MW=80.4100.180 GeV/c2.