Data collected by the Pierre Auger Observatory provide evidence for anisotropy in the arrival directions of the cosmic rays with the highest energies, which are correlated with the positions of relatively nearby active galactic nuclei (AGN). The correlation has maximum significance for cosmic rays with energy greater than ~ 6x10^19 eV and AGN at a distance less than ~ 75 Mpc. We have confirmed the anisotropy at a confidence level of more than 99% through a test with parameters specified a priori, using an independent data set. The observed correlation is compatible with the hypothesis that cosmic rays with the highest energies originate from extra-galactic sources close enough so that their flux is not significantly attenuated by interaction with the cosmic background radiation (the Greisen-Zatsepin-Kuz'min effect). The angular scale of the correlation observed is a few degrees, which suggests a predominantly light composition unless the magnetic fields are very weak outside the thin disk of our galaxy. Our present data do not identify AGN as the sources of cosmic rays unambiguously, and other candidate sources which are distributed as nearby AGN are not ruled out. We discuss the prospect of unequivocal identification of individual sources of the highest-energy cosmic rays within a few years of continued operation of the Pierre Auger Observatory.