  In this paper we consider a superprocess being a measure-valued diffusion
corresponding to the equation $u_{t}=Lu+\alpha u-\beta u^{2}$, where $L$ is the
infinitesimal operator of the \emph{Ornstein-Uhlenbeck process} and
$\beta>0,\:\alpha>0$. The latter condition implies that the process is
\emph{supercritical,} i.e. its total mass grows exponentially. This system is
known to fulfill a law of large numbers. In the paper we prove the
corresponding \emph{central limit theorem}. The limit and the CLT normalization
fall into three qualitatively different classes. In what we call the small
growth rate case the situation resembles the classical CLT. The weak limit is
Gaussian and the normalization is the square root of the size of the system. In
the critical case the limit is still Gaussian, however the normalization
requires an additional term. Finally, when the growth rate is large the
situation is completely different. The limit is no longer Gaussian, the
normalization is substantially larger than the classical one and the
convergence holds in probability. These different regimes arise as a result of
"competition" between spatial smoothing due to the particles' movement and the
system's growth which is local.
