  Perturbative computations of the expectation value of the Wilson loop in N=4
supersymmetric Yang-Mills theory are reported. For the two special cases of a
circular loop and a pair of anti-parallel lines, it is shown that the sum of an
infinite class of ladder-like planar diagrams, when extrapolated to strong
coupling, produces an expectation value characteristic of the results of the
AdS/CFT correspondence, $<W>\sim\exp((constant)\sqrt{g^2N})$. For the case of
the circular loop, the sum is obtained analytically for all values of the
coupling. In this case, the constant factor in front of $\sqrt{g^2N}$ also
agrees with the supergravity results. We speculate that the sum of diagrams
without internal vertices is exact and support this conjecture by showing that
the leading corrections to the ladder diagrams cancel identically in four
dimensions. We also show that, for arbitrary smooth loops, the ultraviolet
divergences cancel to order $g^4N^2$.
