  The vacuum energy for a massless conformally coupled scalar field in a brane
world set up, corresponding to de Sitter branes in a bulk anti-de Sitter
spacetime, is calculated. We use the Euclidean version of the metric which can
be conformally related to a metric similar in form to the Einstein universe
(S^4 times R). Employing zeta-function regularisation we evaluate the one-loop
effective potential and show that the vacuum energy is zero for the one brane
and non-zero for the two brane configuration. We comment on the back-reaction
of this Casimir energy and on the inclusion of a mass term or non-conformal
coupling.
