Recent measurements of stellar orbits provide compelling evidence that the compact radio source Sagittarius A* (refs 4, 5) at the Galactic Centre is a 3.6-million-solar-mass black hole. Sgr A* is remarkably faint in all wavebands other than the radio region, however, which challenges current theories of matter accretion and radiation surrounding black holes. The black hole's rotation rate is not known, and therefore neither is the structure of space-time around it. Here we report high-resolution infrared observations of Sgr A* that reveal `quiescent' emission and several flares. The infrared emission originates from within a few milliarcseconds of the black hole, and traces very energetic electrons or moderately hot gas within the innermost accretion region. Two flares exhibit a 17-minute quasi-periodic variability. If the periodicity arises from relativistic modulation of orbiting gas, the emission must come from just outside the event horizon, and the black hole must be rotating at about half of the maximum possible rate.