We determine the CKM matrix element, |Vcb|, by measuring the differential decay rate of B → D*+ℓv as a function of w = vB · νD*. We extrapolate the decay rate to the kinematic point, w = 1, where the form factor, F(w), is accurately predicted by theory. We find f(1)|vcb| = 0.0424 ± 0.0018(stat.) ± 0.0019(syst.). Integration of the differential decay rate, dT/dw, yields the B → D*+ℓv branching fraction, B(B → D*+ℓv = (5.66 ± 0.29 ± 0.33)%. The data are derived from 3.3 million BB events collected by the CLEO detector at CESR.