The statistics of energy levels of a rectangular billiard that is perturbed by a strong localized potential are studied analytically and numerically, when this perturbation is at the center or at a typical position. Different results are found for these two types of position. If the scatterer is at the center, the symmetry leads to additional contributions, some of which are related to the angular dependence of the potential. The limit of the delta-like scatterer is obtained explicitly. The form factor, which is the Fourier transform of the energy-energy correlation function, is calculated analytically, in the framework of the semiclassical geometrical theory of diffraction, and numerically. Contributions of classical orbits that are nondiagonal are calculated and are found to be essential.