  We report on a non-perturbative evaluation of the renormalization factors for
the vector and axial-vector currents, $Z_V$ and $Z_A$, in the quenched
domain-wall QCD (DWQCD) with plaquette and renormalization group improved gauge
actions. We take the Dirichlet boundary condition for both gauge and
domain-wall fermion fields on the finite box, and introduce the flavor-chiral
Ward-Takahashi identities to calculate the renormalization factors. As a test
of the method, we numerically confirm the expected relation that $Z_V \simeq
Z_A$ in DWQCD. Employing two different box sizes for the numerical simulations
at several values of the gauge coupling constant $g^2$ and the domain-wall
height $M$, we extrapolate $Z_V$ to the infinite volume to remove $a/L$ errors.
We finally give the interpolation formula of $Z_V$ in the infinite volume as a
function of $g^2$ and $M$.
