We investigate a fully frustrated Ising model with four nearest-neighbor interaction constants on a square lattice. We obtain an exact form of the partition function by using a decimation transformation and a dual transformation. The partition function is then expressed as a product of two partition functions each of which is the one of the Ising model with a nearest-neighbor interaction constant on a square lattice. Special systems with two interaction constants are studied in detail and phase transitions for these systems are discussed.