We present a study of B-0 -> D-s*(+)pi(-) and B-0 -> D-s*K--(+) decays based on a sample of 657 x 10(6) B (B) over bar events collected with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider. We measure the branching fractions to be B(B-0 -> D-s*(+)pi(-)) = (1.75 +/- 0.34(stat) +/- 0.17(syst) +/- 0.11(B)) x 10(-5) and B(B-0 -> D-s*K--(+)) = (2.02 +/- 0.33(stat) +/- 0.18(syst) +/- 0.13(B)) x 10(-5), with significances of 6.1 and 8.0 standard deviations, respectively. The first uncertainty is statistical, the second is due to the experimental systematics, and the third is from uncertainties in the D-s(+) decay branching fractions. From our measurements, we obtain the most precise determination so far of R-D*pi, where R-D*pi is the ratio between amplitudes of the doubly Cabibbo-suppressed decay B-0 -> D*(+)pi(-) and the Cabibbo-favored B-0 -> D*(-)pi(+) decay.