  In the last two decades, there has been a significant progress in the
understanding of ergodic properties of white-forced dissipative PDEs. The
previous studies mostly focus on equations posed on bounded domains since they
rely on different compactness properties and the discreteness of the spectrum
of the Laplacian. In the present paper, we consider the damped complex
Ginzburg--Landau equation on the real line driven by a white-in-time noise.
Under the assumption that the noise is sufficiently non-degenerate, we
establish the uniqueness of stationary measure and exponential mixing in the
dual-Lipschitz metric. The proof is based on coupling techniques combined with
a generalization of Foia\c{s}--Prodi estimate to the case of the real line and
special space-time weighted estimates which help to handle the behavior of
solutions at infinity.
