Single-crystal X-ray structural determinations for the Mg35.12(4)Cu24Ga53.58(6) and Mg35.6(4)Cu24Ga52.66(6) refined compositions (Fd3̄m, Z = 4) reveal empty (Cu,Ga)12 icosahedra and centered MgGa16 icosioctahedra that are interconnected at every vertex to a compact three-dimensional anion network. A small range of variable occupancy exists on one of three Ga and one of four Mg positions. The clusters are well-bonded and held in different sized cavities, in which they are also directly bonded to a Mg cation network. The two networks thus interpenetrate each other, and there are no spacers. The new phase is isostructural with K39In80, K17In41, and the electron-poorer Na35Cd24Ga56, all of which contain clathrate-II-type cation frameworks. Electron counting using the classic (MO-based) cluster assignments indicates that the refined structure is substantially ideal and closed-shell. The symmetry of the present structure does not suggest a ready conversion to an icosahedral quasicrystal or its approximants.