  Renormalization constants of vector ($Z_V$) and axial-vector ($Z_A$) currents
are determined non-perturbatively in quenched QCD for a renormalization group
improved gauge action and a tadpole improved clover quark action using the
Schr\"odinger functional method. Non-perturbative values of $Z_V$ and $Z_A$
turn out to be smaller than the one-loop perturbative values by $O(10%)$ at
$a^{-1}\approx 1$ GeV. A sizable scaling violation of meson decay constants
$f_\pi$ and $f_\rho$ observed with the one-loop renormalization factors remains
even with non-perturbative renormalization.
