The lowest radiative correction to the Casimir energy density between
two parallel plates is calculated using effective field theory. Since
the correlators of the electromagnetic field diverge near the plates,
the regularized energy density is also divergent. However, the regular
ized integral of the energy density is finite and varies with the plat
e separation L as 1/L-7. This apparently paradoxical situation is anal
yzed in an equivalent but more transparent theory of a massless scalar
field in 1 + 1 dimensions confined to a line element of length L and
satisfying Dirichlet boundary conditions.