  We examine the partition function of N=2* supersymmetric SU(N) Yang-Mills
theory on the four-sphere in the large radius limit. We point out that the
large radius partition function, at fixed N, is computed by saddle points lying
on particular walls of marginal stability on the Coulomb branch of the theory
on R^4. For N an even (odd) integer and \theta_YM=0, (\pi), these include a
point of maximal degeneration of the Donagi-Witten curve to a torus where BPS
dyons with electric charge [N/2] become massless. We argue that the dyon
singularity is the lone saddle point in the SU(2) theory, while for SU(N) with
N>2, we characterize potentially competing saddle points by obtaining the
relations between the Seiberg-Witten periods at such points. Using Nekrasov's
instanton partition function, we solve for the maximally degenerate saddle
point and obtain its free energy as a function of g_YM and N, and show that the
results are "large-N exact". In the large-N theory our results provide
analytical expressions for the periods/eigenvalues at the maximally degenerate
saddle point, precisely matching previously known formulae following from the
correspondence between N=2* theory and the elliptic Calogero-Moser integrable
model. The maximally singular point ceases to be a saddle point of the
partition function above a critical value of the coupling, in agreement with
the recent findings of Russo and Zarembo.
