  We investigate an exactly marginal N=1 supersymmetric deformation of SU(N)
N=4 supersymmetric Yang-Mills theory discovered by Leigh and Strassler. We use
a matrix model to compute the exact superpotential for a further massive
deformation of the U(N) Leigh-Strassler theory. We then show how the exact
superpotential and eigenvalue spectrum for the SU(N) theory follows by a
process of integrating-in. We find that different vacua are related by an
action of the SL(2,Z) modular group on the bare couplings of the theory
extending the action of electric-magnetic duality away from the N=4 theory. We
perform non-trivial tests of the matrix model results against semiclassical
field theory analysis. We also show that there are interesting points in
parameter space where condensates can diverge and vacua disappear. Based on the
matrix model results, we propose an exact elliptic superpotential to describe
the theory compactified on a circle of finite radius.
