The new clathrate-I phase Ba8RhxSi46-x-y square y (2.36 < x < 2.74; y = 0.65 for x = 2.74), Pearson symbol cP54, space group Pm (3) over barn, was prepared as a single phase and characterized. For these compositions, the lattice parameter does not change significantly and was found to be a = 10.347(1) angstrom. Phase relations were investigated by means of metallographic and thermal analyses. The incongruently melting phase Ba8RhxSi46-x-y square y was obtained by annealing a solidified melt of the stoichiometric composition at 1000 degrees C for five days. Below 850 degrees C, the clathrate phase slowly decomposes to BaSi2, Si and the new compound BaRh2Si9. Crystal structure and vacancy concentration were investigated by means of single crystal and powder X-ray diffraction and wavelength dispersive X-ray spectroscopy analyses. Quantum chemical calculations show that the Rh 4d states hybridize with Si 3p states so that the DOS structure cannot be simply deduced in a rigid band approach from that of the binary Ba8Si46. Analysis of the chemical bonding applying the ELI-D approach yielded covalent two-center two-electron bonds between Rh and Si atoms. For the composition Ba8Rh2.74Si42.61 square(0.65), a temperature dependent diamagnetic susceptibility and an almost temperature independent electrical resistivity (rho approximate to 5 mu Omega m) were observed. The magnitude of the negative Seebeck coefficient is increasing linearly with temperature from 0 mu V K-1 at 2 K to -35 mu V K-1 at 850 K indicating n-type electrical conductivity.