  A new approximation of the cluster variational method is introduced for the
three-dimensional Ising model on the simple cubic lattice. The maximal cluster
is, as far as we know, the largest ever used in this method. A message-passing
algorithm, generalized belief propagation, is used to minimize the variational
free energy. Convergence properties and performance of the algorithm are
investigated. The approximation is used to compute the spontaneous
magnetization, which is then compared to previous results. Using the present
results as the last step in a sequence of three cluster variational
approximations, an extrapolation is obtained which captures the leading
critical behaviour with a good accuracy.
