  The Casimir energy of a dilute homogeneous nonmagnetic dielectric ball at
zero temperature is derived analytically for the first time for an arbitrary
physically possible frequency dispersion of dielectric permittivity
$\epsilon(i\omega)$. A microscopic model of dielectrics is considered,
divergences are absent in calculations because an average interatomic distance
$\lambda$ is a {\it physical} cut-off in the theory. This fact has been
overlooked before, which led to divergences in various macroscopic approaches
to the Casimir energy of connected dielectrics.
