Order a improved renormalization constants

We present nonperturbative results for the constants needed for on-shell O(a) improvement of bilinear operators composed of Wilson fermions. We work at {beta}=6.0 and 6.2 in the quenched approximation. The calculation is done by imposing axial and vector Ward identities on correlators similar to those used in standard hadron mass calculations. A crucial feature of the calculation is the use of nondegenerate quarks. We also obtain results for the constants needed for off-shell O(a) improvement of bilinears, and for the scale- and scheme-independent renormalization constants, Z{sub A}, Z{sub V}, and Z{sub S}/Z{sub P}. Several of the constants are determined using a variety of different Ward identities, and we compare their relative efficacies. In this way, we find a method for calculating c{sub V} that gives smaller errors than that used previously. Wherever possible, we compare our results with those of the ALPHA Collaboration (who use the Schro''dinger functional) and with one-loop tadpole-improved perturbation theory.