Owing to its extremely high luminosity and long duration, supernova (SN) 2006gy radiated more energy in visual light than any other known SN. Two hypotheses to explain its high luminosity at early times—that it was powered by shock interaction with circumstellar material (CSM) as implied by its Type IIn spectrum, or that it was fueled by radioactive decay from a large mass of $^56$Ni synthesized in a pair-instability SN—predicted different late-time properties. Here we present observations of SN 2006gy obtained more than a year after discovery. We were unable to detect it at visual wavelengths, but clear near-infrared (IR) K$^′$ and H-band detections show that it is still at least as luminous as the peak of a normal Type II SN. We also present spectra giving an upper limit to the late-time Hα luminosity of lesssim10$^39$ erg s$^-1$. Based on the weak late-time Hα, X-ray, and radio emission, combined with the difficulty of explaining the shift to IR wavelengths, we can rule out ongoing CSM interaction as the primary late-time power source of SN 2006gy. Instead, we propose that the evolution of SN 2006gy is consistent with one of two possible scenarios: (1) a pair-instability SN plus modest CSM interaction, where the radioactive decay luminosity shifts to the IR because of dust formation; or (2) an IR echo, where radiation emitted during peak luminosity heats a pre-existing dust shell at radii near 1 light year, requiring the progenitor star to have ejected another shell of åisebox-0.5ex 10 M$_☉$ about 1500 yr before the SN.