  We compute the leading radiative correction to the Casimir force between two
parallel plates in the $\lambda\Phi^4$ theory. Dirichlet and periodic boundary
conditions are considered. A heuristic approach, in which the Casimir energy is
computed as the sum of one-loop corrected zero-point energies, is shown to
yield incorrect results, but we show how to amend it. The technique is then
used in the case of periodic boundary conditions to construct a perturbative
expansion which is free of infrared singularities in the massless limit. In
this case we also compute the next-to-leading order radiative correction, which
turns out to be proportional to $\lambda^{3/2}$.
