We present a measurement of the mass of the W boson, the charged mediator of the weak force. The measurement is performed using data corresponding to 8.8/fb of integrated luminosity collected by the CDF detector at the Fermilab Tevatron between 2002 and 2011. In a sample with approximately 4 million W boson candidate events, we measure M(W) = 80433.5+-6.4(stat)+-6.9(syst) MeV = 80433.5+-9.4 MeV. The measurement has a precision exceeding that of all previous measurements of M(W) combined and it is in significant tension with the standard model expectation.