  We compute the coefficients of an infinite family of chiral primary operators
in the local operator expansion of a circular Wilson loop in N=4 supersymmetric
Yang-Mills theory. The computation sums all planar rainbow Feynman graphs. We
argue that radiative corrections from planar graphs with internal vertices
cancel in leading orders and we conjecture that they cancel to all orders in
perturbation theory. The coefficients are non-trivial functions of the 'tHooft
coupling and their strong coupling limits are in exact agreement with those
previously computed using the AdS/CFT correspondence. They predict the
subleading orders in strong coupling and could in principle be compared with
string theory calculations.
