We present a measurement of the W-boson mass using 200 pb(-1) of data collected in p (p) over bar collisions at root s=1.96 TeV by the CDF II detector at run II of the Fermilab Tevatron. With a sample of 63 964 W -> e nu candidates and 51 128 W ->mu nu candidates, we measure M(W)=80413 +/- 34(stat) +/- 34 (syst) = 80 413 +/- 48 MeV/c(2). This is the most precise single measurement of the W-boson mass to date.