  String theory implies that field theories containing gravity are in a certain
sense `products' of gauge theories. We make this product structure explicit up
to two loops for the relatively simple case of N=8 supergravity four-point
amplitudes, demonstrating that they are `squares' of N=4 super-Yang-Mills
amplitudes. This is accomplished by obtaining an explicit expression for the
$D$-dimensional two-loop contribution to the four-particle S-matrix for N=8
supergravity, which we compare to the corresponding N=4 Yang-Mills result. From
these expressions we also obtain the two-loop ultraviolet divergences in
dimensions D=7 through D=11. The analysis relies on the unitarity cuts of the
two theories, many of which can be recycled from a one-loop computation. The
two-particle cuts, which may be iterated to all loop orders, suggest that
squaring relations between the two theories exist at any loop order. The
loop-momentum power-counting implied by our two-particle cut analysis indicates
that in four dimensions the first four-point divergence in N=8 supergravity
should appear at five loops, contrary to the earlier expectation, based on
superspace arguments, of a three-loop counterterm.
