The CDF and D0 Collaborations obtain new measurements of the W boson mass using data taken between 1994 and 1996. CDF updates the previous result obtained from W -> mu nu decays, reducing many systematic uncertainties, and performs a new measurement using a high statistics sample of W -> e nu events; combining these measurements with the previous ones yields M^CDF_W = 80.433 +/- 0.079 GeV/c^2. D0 also extends the previous result, based on central electron events, by including events with electrons landed in the forward calorimeters; the combined D0 result is M^D0_W = 80.474 +/- 0.093 GeV/c^2. The numbers obtained by the two experiments can be combined, together with the older UA2 one (M^UA2_W = 80.36 +/- 0.37 GeV/c^2), by assuming a 25 MeV/c^2 common error, to yield a hadron collider average of M_W = 80.448 +/- 0.062 GeV/c^2.