We have measured the W-boson mass M(W) using data corresponding to 2.2 fb(-1) of integrated luminosity collected in pp 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/c2. This is the most precise measurement of the W-boson mass to date and significantly exceeds the precision of all previous measurements combined.