We describe a measurement of the W boson mass mW using 200  pb−1 of s√=1.96  TeV pp collision data taken with the CDF II detector. With a sample of 63 964 W→eν candidates and 51 128 W→μν candidates, we measure mW=[80.413±0.034(stat)±0.034(sys)=80.413±0.048]  GeV/c2. This is the single most precise mW measurement to date. When combined with other measured electroweak parameters, this result further constrains the properties of new unobserved particles coupling to W and Z bosons.