  The Casimir energy corresponding to a massive scalar field with Dirichlet
boundary conditions on a spherical bag is obtained. The field is considered,
separately, inside and outside the bag. The renormalization procedure that is
necessary to apply in each situation is studied in detail, in particular the
differences occurring with respect to the case when the field occupies the
whole space. The final result contains several constants that experience
renormalization and can be determined only experimentally. The non-trivial
finite parts that appear in the massive case are found exactly, providing a
precise determination of the complete, renormalized zero-point energy for the
first time.
