  We prove the formula for the complete tree-level $S$-matrix of
$\mathcal{N}=8$ supergravity recently conjectured by two of the authors. The
proof proceeds by showing that the new formula satisfies the same BCFW
recursion relations that physical amplitudes are known to satisfy, with the
same initial conditions. As part of the proof, the behavior of the new formula
under large BCFW deformations is studied. An unexpected bonus of the analysis
is a very straightforward proof of the enigmatic $1/z^2$ behavior of gravity.
In addition, we provide a description of gravity amplitudes as a
multidimensional contour integral over a Grassmannian. The Grassmannian
formulation has a very simple structure; in the N$^{k-2}$MHV sector the
integrand is essentially the product of that of an MHV and an $\overline{{\rm
MHV}}$ amplitude, with $k+1$ and $n-k-1$ particles respectively.
