We present a measurement of the W-boson mass, M-W, using data corresponding to 2.2 fb(-1) of integrated luminosity collected in p (p) over bar collisions at root s = 1.96 TeV with the CDF II detector at the Fermilab Tevatron. The selected sample of 470 126 W -> ev candidates and 624 708 W -> mu nu candidates yields the measurement M-W = 80387 +/- 12 (stat) +/- 15(syst) = 80387 +/- 19 MeV/c(2). This is the most precise single measurement of the W-boson mass to date.