  Recently it was conjectured that a certain infinite-dimensional "diagonal"
subgroup of BMS supertranslations acting on past and future null infinity
(${\mathscr I}^-$ and ${\mathscr I}^+$) is an exact symmetry of the quantum
gravity ${\cal S}$-matrix, and an associated Ward identity was derived. In this
paper we show that this supertranslation Ward identity is precisely equivalent
to Weinberg's soft graviton theorem. Along the way we construct the canonical
generators of supertranslations at ${\mathscr I}^\pm$, including the relevant
soft graviton contributions. Boundary conditions at the past and future of
${\mathscr I}^\pm$ and a correspondingly modified Dirac bracket are required.
The soft gravitons enter as boundary modes and are manifestly the Goldstone
bosons of spontaneously broken supertranslation invariance.
