Strong interactions between electrons in a solid material can lead to surprising properties. A prime example is the Mott insulator, in which suppression of conductivity occurs as a result of interactions rather than a filled Bloch band. Proximity to the Mott insulating phase in fermionic systems is the origin of many intriguing phenomena in condensed matter physics, most notably high-temperature superconductivity. The Hubbard model, which encompasses the essential physics of the Mott insulator, also applies to quantum gases trapped in an optical lattice. It is therefore now possible to access this regime with tools developed in atomic physics. However, an atomic Mott insulator has so far been realized only with a gas of bosons, which lack the rich and peculiar nature of fermions. Here we report the formation of a Mott insulator of a repulsively interacting two-component Fermi gas in an optical lattice. It is identified by three features: a drastic suppression of doubly occupied lattice sites, a strong reduction of the compressibility inferred from the response of double occupancy to an increase in atom number, and the appearance of a gapped mode in the excitation spectrum. Direct control of the interaction strength allows us to compare the Mott insulating regime and the non-interacting regime without changing tunnel-coupling or confinement. Our results pave the way for further studies of the Mott insulator, including spin-ordering and ultimately the question of d-wave superfluidity.