  In this paper we consider a branching particle system consisting of particles
moving according to the Ornstein-Uhlenbeck process in $\Rd$ and undergoing a
binary, supercritical branching with a constant rate $\lambda>0$. This system
is known to fulfil a law of large numbers (under exponential scaling). In the
paper we prove the corresponding central limit theorem. The limit and the CLT
normalisation fall into three qualitatively different classes. In, what we
call, the small branching rate case the situation resembles the classical one.
The weak limit is Gaussian and normalisation is the square root of the size of
the system. In the critical case the limit is still Gaussian, however the
normalisation requires an additional term. Finally, when branching has large
rate the situation is completely different. The limit is no longer Gaussian,
the normalisation is substantially larger than the classical one and the
convergence holds in probability.
  We prove also that the spatial fluctuations are asymptotically independent of
the fluctuations of the total number of particles (which is a Galton-Watson
process).
