The CDF collaboration has measured the mass of the W boson to be M_V = 80.41 \pm 0.18 GeV/c^2 using 5718 W -> e nu events and 3268 W -> mu nu events collected in approximately 19 pb^-1 from the 1992-93 collider run at the Fermilab Tevatron. This measurement has an uncertainty half that of the best previously published measurements.