  The one-dimensional Ising model is easily generalized to a \textit{genuinely
nonequilibrium} system by coupling alternating spins to two thermal baths at
different temperatures. Here, we investigate the full time dependence of this
system. In particular, we obtain the evolution of the magnetisation, starting
with arbitrary initial conditions. For slightly less general initial
conditions, we compute the time dependence of all correlation functions, and
so, the probability distribution. Novel properties, such as oscillatory decays
into the steady state, are presented. Finally, we comment on the relationship
to a reaction-diffusion model with pair annihilation and creation.
