  In this article, we investigate the asymptotic behavior of the solution to a
one-dimensional stochastic heat equation with random nonlinear term generated
by a stationary, ergodic random field. We extend the well-known central limit
theorem for finite-dimensional diffusions in random environment to this
infinite-dimensional setting. Due to our result, a central limit theorem in
$L^1$ sense with respect to the randomness of the environment holds under a
diffusive time scaling. The limit distribution is a centered Gaussian law whose
covariance operator is explicitly described. It concentrates only on the space
of constant functions.
