  The invariant measure of a one-dimensional Allen-Cahn equation with an
additive space-time white noise is studied. This measure is absolutely
continuous with respect to a Brownian bridge with a density which can be
interpreted as a potential energy term. We consider the sharp interface limit
in this setup. In the right scaling this corresponds to a Gibbs type measure on
a growing interval with decreasing temperature. Our main result is that in the
limit we still see exponential convergence towards a curve of minimizers of the
energy if the interval does not grow too fast. In the original scaling the
limit measure is concentrated on configurations with precisely one jump. This
jump is distributed uniformly.
