We examine the exponential decay law for an excited quantum system in the case where the state amplitude has a “memory,” that is, where the rate at which the amplitude is decaying at a particular time depends to some extent on values of the amplitude at previous times. For a particular choice of memory, characterized by a memory time τ, the amplitude obeys the differential equation of a damped harmonic oscillator. When τ is greater than a certain critical memory time τc, comparable to the natural lifetime of the state, the amplitude becomes an exponentially damped oscillation in time. When τ is less than τc, the decay is nearly exponential with a decay rate somewhat greater than that calculated from standard theory. By comparing theory with experiment, we can place an upper limit on the memory time for the radiative decay of atoms.