We discuss the mass-deformed = 4 SU(N) supersymmetric Yang-Mills theory (also known as the = 1* theory). We analyze how the correlation functions of this theory transform under S-duality, and which correlation functions depend holomorphically on the complexified gauge coupling τ. We provide exact modular-covariant expressions for the vacuum expectation values of chiral operators in the massive vacua of the = 1* theory. We exhibit a novel modular symmetry of the chiral sector of the theory in each vacuum, which acts on the coupling = (pτ+k)/q, where p, k and q are integers which label the different vacua. In the strong coupling limit, we compare our results to the results of Polchinski and Strassler in the string theory dual of this theory, and find non-trivial agreement after operator mixings are taken into account. In particular we find that their results are consistent with the predicted modular symmetry in . Our results imply that certain singularities found in solutions to five-dimensional gauged supergravity should not be resolvable in string theory, since there are no field theory vacua with corresponding vacuum expectation values in the large-N limit.