  In this paper, we investigate the finite connectivity spin-glass problem. Our
work is focused on the expansion around the point of infinite connectivity of
the free energy of a spin glass on a graph with Poissonian distributed
connectivity: we are interested to study the first-order correction to the
infinite connectivity result for large values or the connectivity $z$. The same
calculations for one and two replica symmetry breakings were done in previous
works; the result for the first-order correction was divergent in the limit of
zero temperature and it was suggested that it was an artifact for having a
finite number of replica symmetry breakings. In this paper we are able to
calculate the expansion for an infinite number of replica symmetry breakings:
in the zero-temperature limit, we obtain a well defined free energy. We have
shown that cancellations of divergent terms occur in the case of an infinite
number of replica symmetry breakings and that the pathological behavior of the
expansion was due only to the finite number of replica symmetry breakings.
