A Hanbury Brown and Twiss experiment for a beam of electrons has been realized in a two-dimensional electron gas in the quantum Hall regime. A metallic split gate serves as a tunable beam splitter to partition the incident beam into transmitted and reflected partial beams. In the nonequilibrium case the fluctuations in the partial beams are shown to be fully anticorrelated, demonstrating that fermions exclude each other. In equilibrium, the cross-correlation of current fluctuations at two different contacts is also found to be negative and nonzero, provided that a direct transmission exists between the contacts.