We report the first observation of two Cabibbo-suppressed decay modes of the B-s(0) meson. Using a sample of p (p) over bar collisions at root s = 1.96 TeV corresponding to 5: 9 fb(-1) of integrated luminosity collected with the CDF II, the collider detector at the Fermilab Tevatron, we search for new B-s(0) decay modes in a sample of events containing J/psi -> mu(+)mu(-) decays. We reconstruct a B-s(0) -> J/psi K*(892)(0) 0 signal with K*(892)(0) -> K+ pi(-), observing a yield of 151 +/- 25 events with a statistical significance of 8.0 sigma. We also reconstruct a B-s(0) -> J/psi K-s(0) signal with K-s(0) -> pi(+)pi(-) , observing a yield of 64 +/- 14 events with a statistical significance of 7.2 sigma. From these yields, we extract the branching ratios B(B-s(0) -> J/psi K* (892)(0) = (8.3 +/- 3.8) X 10(-5) and B(B-s(0) -> J/psi K-0) = (3.5 +/- 0.8) X 10(-5), where statistical, systematic, and fragmentation-fraction uncertainties are included in the combined uncertainty.
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