On the Relationship between Yang-Mills Theory and Gravity and its Implication for Ultraviolet Divergences

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.