  We study the mean field dilute model of a ferromagnet. We find and prove an
expression for the free energy density at high temperature, and at temperature
zero. We find the critical line of the model, separating the phase with zero
magnetization from the phase with symmetry breaking. We also compute exactly
the entropy at temperature zero, which is strictly positive. The physical
behavior at temperature zero is very interesting and related to infinite
dimensional percolation, and suggests possible behaviors at generic low
temperatures. Lastly, we provide a complete solution for the annealed model.
Our results hold both for the Poisson and the Bernoulli versions of the model.
