Here we report on simulations which, for the first time, simultaneously track star formation and black hole growth, and associated feedback processes, during collisions of galaxies. We show that encounters and mergers of galaxies lead to strong nuclear gas inflows, fueling both powerful starbursts and rapid growth of central black holes. Energetic feedback from accretion onto black holes eventually expels sufficient amount of gas from the merger remnant to quench star formation as well as further black hole growth. At the completion of such an event, the stellar velocity dispersion of the remnant spheroid correlates with the final black hole mass in the manner indicated by observations of the MBH − σ relation.