Mixed S2-/S1- oxidation states have been,discovered in the new quaternary compound NaBa2Cu3S5. Synthesized from the reaction of Cu in a molten alkali metal/polysulfide flux, the compound crystallizes in monoclinic space group C2/m with a = 16.5363(7) angstrom, b = 5.5374(5) degrees, c = 10.3717(10) angstrom, beta = 98.535(8)degrees. The Na+ Ba-2(+2) [Cu3+S3]S-3-(2)2- crystal structure contains layers of edge sharing CuS4 tetrahedra and sheets of S-2(2-) dimers. These layers are separated by mixed Ba/Na cation layers. The conductivity of the single crystals of NaBa2Cu3S5 is similar to 450 S cm(-1) at room temperature, and increasing conductivity with decreasing temperature is observed, indicating metallic behavior despite the optical band gap of 0.45 eV. A small positive thermopower (45-55 mu V K-1 from 300 K to 500 K) and Hall effect measurements also confirm p-type conductivity with carrier concentration at 200 K of similar to 1.6 X 10(21) cm(-3) and a hole mobility of similar to 2 cm(2) V-1 s(-1). NaBa2Cu3S5 exhibits temperature-independent Pauli paramagnetism.