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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 W→eν events recorded in an integrated luminosity of 84 pb-1 gives a mass Mw=80.473±0.065(stat) ±0.092(syst) GeVc2. A fit to the transverse mass spectrum of a sample of 14740 W→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 systematic uncertainties are the uncertainties in the electron energy scale and the muon momentum scale, 0.075 GeV/c2 and 0.085 GeV/c2, respectively. The combined value for the electron and muon channel is Mw=80.470±0.089 GeV/c2. When combined with previously published CDF measurements, we obtain Mw=80.433±0.079 GeV/c2.