  We present an expression for the leading-color (planar) four-loop four-point
amplitude of N=4 supersymmetric Yang-Mills theory in 4-2 e dimensions, in terms
of eight separate integrals. The expression is based on consistency of
unitarity cuts and infrared divergences. We expand the integrals around e=0,
and obtain analytic expressions for the poles from 1/e^8 through 1/e^4. We give
numerical results for the coefficients of the 1/e^3 and 1/e^2 poles. These
results all match the known exponentiated structure of the infrared
divergences, at four separate kinematic points. The value of the 1/e^2
coefficient allows us to test a conjecture of Eden and Staudacher for the
four-loop cusp (soft) anomalous dimension. We find that the conjecture is
incorrect, although our numerical results suggest that a simple modification of
the expression, flipping the sign of the term containing zeta_3^2, may yield
the correct answer. Our numerical value can be used, in a scheme proposed by
Kotikov, Lipatov and Velizhanin, to estimate the two constants in the
strong-coupling expansion of the cusp anomalous dimension that are known from
string theory. The estimate works to 2.6% and 5% accuracy, providing
non-trivial evidence in support of the AdS/CFT correspondence. We also use the
known constants in the strong-coupling expansion as additional input to provide
approximations to the cusp anomalous dimension which should be accurate to
under one percent for all values of the coupling. When the evaluations of the
integrals are completed through the finite terms, it will be possible to test
the iterative, exponentiated structure of the finite terms in the four-loop
four-point amplitude, which was uncovered earlier at two and three loops.
