We present a measurement of the W boson mass using data collected with the CDF detector during the 1994–1995 collider run at the Fermilab Tevatron. A fit to the transverse mass spectrum of a sample of 30115 W⃗ eν events recorded in an integrated luminosity of 84 pb-1 gives a mass MW=80.473±0.065(stat)±0.092(syst)GeV/c2. A fit to the transverse mass spectrum of a sample of 14740 W⃗ μν events from 80 pb-1 gives a mass MW=80.465±0.100(stat)±0.103(syst)GeV/c2. The dominant contributions to the sys- tematic uncertainties are the uncertainties in the electron energy scale and the muon momentum scale, 0.075GeV/c2 and 0.085GeV/c2, respectively. The combined value for the electron and muon channel is MW=80.470±0.089GeV/c2. When combined with previously published CDF measurements, we obtain MW=80.433±0.079GeV/c2.