  We consider invariant measures for the stochastic Burgers equation on
$\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian
noise that is white in time and smooth in space. An invariant measure is
indecomposable, or extremal, if it cannot be represented as a convex
combination of other invariant measures. We show that for each
$a\in\mathbb{R}$, there is a unique indecomposable law of a
spacetime-stationary solution with mean $a$, in a suitable function space. We
also show that solutions starting from spatially-decaying perturbations of
mean-$a$ periodic functions converge in law to the extremal space-time
stationary solution with mean $a$ as time goes to infinity.
