The light curves of Type Ia supernovae are routinely used to constrain cosmology models. Driven by radioactive decay of 56Ni, the light curves steadily decline over time, but >150 days past explosion, the near-infrared portion is poorly characterized. We report a year-long plateau in the near-infrared light curve at 150-500 days, followed by a second decline phase accompanied by a possible appearance of [Fe I] emission lines. This near-infrared plateau contrasts sharply with Type IIP plateaus and requires a new physical mechanism. We suggest a such as masking of the "near-infrared catastrophe," a predicted yet unobserved sharp light-curve decline, by scattering of ultraviolet photons to longer wavelengths. The transition off the plateau could be due to a change in the dominant ionization state of the supernova ejecta. Our results shed new light on the complex radiative transfer processes that take place in Type Ia supernovae and enhance their use as "standard candles."