  We consider a two-level system coupled to an environment that evolves
non-adiabatically. We present a non-perturbative method for determining the
persistence amplitude whose phase contains all the corrections to Berry's phase
produced by the non-adiabatic motion of the environment. Specifically, it
includes the effects of transitions between the two energy levels to all orders
in the non-adiabatic coupling. The problem of determining all non-adiabatic
corrections is reduced to solving an ordinary differential equation to which
numerical methods should provide solutions in a variety of situations. We apply
our method to a particular example that can be realized as a magnetic resonance
experiment, thus raising the possibility of testing our results in the lab.
