  In a certain kinematic limit, where the effects of spacetime curvature (and
other background fields) greatly simplify, the light-cone gauge world-sheet
action for a type IIB superstring on AdS_5 x S^5 reduces to that of a free
field theory. It has been conjectured by Berenstein, Maldacena, and Nastase
that the energy spectrum of this string theory matches the dimensions of
operators in the appropriately defined large R-charge large-N_c sector of N=4
supersymmetric Yang--Mills theory in four dimensions. This holographic
equivalence is thought to be exact, independent of any simplifying kinematic
limits. As a step toward verifying this larger conjecture, we have computed the
complete set of first curvature corrections to the spectrum of light-cone gauge
string theory that arises in the expansion of AdS_5 x S^5 about the plane-wave
limit. The resulting spectrum has the complete dependence on lambda = g_YM^2
N_c; corresponding results in the gauge theory are known only to second order
in lambda. We find precise agreement to this order, including the N=4 extended
supermultiplet structure. In the process, we demonstrate that the complicated
schemes put forward in recent years for defining the Green--Schwarz superstring
action in background Ramond-Ramond fields can be reduced to a practical (and
correct) method for quantizing the string.
