  In the \beta-deformed N=4 supersymmetric SU(N) Yang-Mills theory we study the
class of operators O_J = Tr(\Phi_i^J \Phi_k), i\neq k and compute their exact
anomalous dimensions for N,J\to\infty. This leads to a prediction for the
masses of the corresponding states in the dual string theory sector. We test
the exact formula perturbatively up to two loops. The consistency of the
perturbative calculation with the exact result indicates that in the planar
limit the one--loop condition g^2=h\bar{h} for superconformal invariance is
indeed sufficient to insure the {\em exact} superconformal invariance of the
theory. We present a direct proof of this point in perturbation theory. The O_J
sector of this theory shares many similarities with the BMN sector of the N=4
theory in the large R--charge limit.
