In the context of an idealized model describing an atom coupled to black-body radiation at a sufficiently high positive temperature, we show that the atom will end up being ionized in the limit of large times. Mathematically, this is translated into the statement that the coupled system does not have any time-translation invariant state of positive (asymptotic) temperature, and that the expectation value of an arbitrary finite-dimensional projection in an arbitrary initial state of positive (asymptotic) temperature tends to zero, as time tends to infinity. These results are formulated within the general framework of W ^*-dynamical systems, and the proofs are based on Mourre's theory of positive commutators and a new virial theorem. Results on the so-called standard form of a von Neumann algebra play an important role in our analysis.