  In a {\cal N}=1 superspace formulation of {\cal N}=4 Yang-Mills theory we
obtain the anomalous dimensions of chiral operators with large R charge J \to
\infty keeping g^2 N/J^2 finite, to all orders of perturbation theory in the
planar limit. Our result proves the conjecture that the anomalous dimensions
are indeed finite in the above limit. This amounts to an exact check of the
proposed duality between a sector of {\cal N}=4 Yang-Mills theory with large R
charge J and string theory in a pp-wave background.
