Phase relations and solidification behavior in the Ge-rich part of the phase diagram have been determined in two isothermal sections at 700 and 750 degrees C and in a liquidus projection. A reaction scheme has been derived in the form of a Schulz-Scheil diagram. Phase equilibria are characterized by three ternary compounds: tau(1)-BaRhGe3 (BaNiSn3-type) and two novel phases, tau(2)-Ba3Rh4Ge16 and tau(3)-Ba5Rh15Ge36-x, both forming in peritectic reactions. The crystal structures of tau(2) and tau(3) have been elucidated from single-crystal X-ray intensity data and were found to crystallize hi unique structure types: Ba3Rh4Ge16 is tetragonal (I4/mmm, a = 0.65643(2) nm, c = 2.20367(8) nm, and R-F = 0.0273), whereas atoms in Ba5Rh15Ge36-x (x = 0 25) arrange in a large orthorhombic unit cell (Fddd, a = 0.84570(2) nm, b = 1.4725(2) nm, c = 6.644(3) nm, and R-F = 0.034). The body-centered-cubic superstructure of binary Ba8Ge43 square(3) was observed to extend at 800 degrees C to Ba8Rh0.6Ge43 square(2.4), while the clathrate type I phase, kappa(I)-Ba8RhxGe46-x-y square(y), reveals a maximum solubility of x = 1.2 Rh atoms in the structure at a vacancy level of y = 2.0. The cubic lattice parameter increases with increasing Rh content. Clathrate I decomposes eutectoidally at 740 degrees C: K-I double left right arrow (Ge) + kappa(IX) + tau(2). A very small solubility range is observed at 750 degrees C for the clathrate IX, kappa(IX)-Ba6RhxGe25-x (x similar to 0.16). Density functional theory calculations have been performed to derive the enthalpies of formation and densities of states for various compositions Ba8RhxGe46-x (x = 0-6). The physical properties have been investigated for the phases kappa(I), tau(1), tau(2), and tau(3), documenting a change from thermoelectric (kappa(1)) to superconducting behavior (tau(2)). The electrical resistivity of kappa(1)-Ba8Rh1.2Ge42.8 square(2.0) increases almost linearly with the temperature from room temperature to 730 K, and the Seebeck coefficient is negative throughout the same temperature range. tau(1)-BaRhGe3 has a typical metallic electrical resistivity. A superconducting transition at T-C = 6.5 K was observed for tau(2)-Ba3Rh4Ge16, whereas tau(3)-Ba5Rh15Ge35.75 showed metallic-like behavior down to 4 K.