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 --> mu nu events recorded in an integrated luminosity of 19.7 pb(-1) gives M(W)(mu) = 80.310 +/- 0.205 (stat) +/- 0.130 (syst) GeV/c(2). A fit to the transverse mass spectrum ofa sample of 5718 W --> ev events recorded in 18.2 pb(-1) gives M(W)(e) = 80.490 +/- 0.145 (stat) +/- 0.175 (syst) GeV/c(2). Combining the muon and electron results, accounting for correlated uncertainties, yields Mw = 80.410 +/- 0.180 GeV/c(2). RI Vucinic, Dejan/C-2406-2008; Chiarelli, Giorgio/E-8953-2012; Azzi, Patrizia/H-5404-2012