We have measured the W-boson mass M_W using data corresponding to 2.2 fb^{-1} of integrated luminosity collected in p\bar{p} collisions at \sqrt{s}=1.96 TeV with the CDF II detector at the Fermilab Tevatron collider. Samples consisting of 470,126 W -> eν candidates and 624,708 W -> µν candidates yield the measurement M_W = 80,387 ± 12_stat ± 15_syst = 80,387 ± 19 MeV/c^2. This is the most precise measurement of the W-boson mass to date.