Using chiral Ward identities, we determine the renormalization constants of bilinear quark operators for the Sheikholeslami-Wohlert action lattice at {beta}=6.2. The results are obtained with a high degree of accuracy. For the vector current renormalization constant we obtain {ital Z}{sub {ital V}}=0.817{plus_minus}2{plus_minus}8, where the first error is statistical and the second is due to mass dependence of {ital Z}{sub {ital V}}. This is close to the perturbative value of 0.83. For the axial current renormalization constant we obtain {ital Z}{sub {ital A}}=1.045{sub {minus}14}{sup +10}, significantly higher than the value obtained in peturbation theory. This is shown to reduce the difference between lattice estimates and the experimental values for the pseudoscalar meson decay constants, but a significant discrepancy remains. The ratio of pseudoscalar to scalar renormalization constants, {ital Z}{sub {ital P}}/{ital Z}{sub {ital S}}, is less well determined, but seems to be slightly lower than the perturbative value.