  We present non-perturbative 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 non-degenerate 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_A), (Z_V) and (Z_S/Z_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 $c_V$ that gives smaller errors than that used
previously. Wherever possible, we compare our results with those of the ALPHA
collaboration (who use the Schr\"odinger functional) and with 1-loop
tadpole-improved perturbation theory.
