  A frustrated Ising model on a diamond hierarchical lattice is studied. We
obtain the exact partition function of this model and calculate the transition
temperature, specific heat, entropy, magnetization, and ferromagnetic
correlation function. Depending on the magnitude of a parameter giving the
frustration, there exist three types of ground states: ferromagnetic, classical
spin-liquid with highly developed short-range order, and paramagnetic. The
dependence of the zero-temperature entropy on the frustration parameter has an
infinite number of steps. The temperature dependence of the specific heat
exhibits many peaks with decreasing temperature and entropy loss. The dominant
spin configurations at low temperatures are also specified.
