It is usually assumed that a massive relic species, which comes to dominate the mass density of the Universe and later decays, ``heats up'' the Universe when the age of the Universe \ensuremath{\simeq} its lifetime. We show that if its decay follows the usual exponential decay law, then the Universe is never reheated, rather it just cools more slowly. We calculate the evolution of the temperature and entropy, and find that to within numerical factors of order unity, the usual estimates for the entropy increase are correct. Our results have implications for primordial nucleosynthesis in scenarios where a massive relic with lifetime \ensuremath{\simeq} ${10}^{\mathrm{\ensuremath{-}}2}$--${10}^{3}$ sec is present, and for baryogenesis in the new inflationary Universe scenario.