  The gravitational $\mathcal{S}$-matrix defined with an infrared (IR) cutoff
factorizes into hard and soft factors. The soft factor is universal and
contains all the IR and collinear divergences. Here we show, in a momentum
space basis, that the intricate expression for the soft factor is fully
reproduced by two boundary currents, which live on the celestial sphere. The
first of these is the supertranslation current, which generates spacetime
supertranslations. The second is its symplectic partner, the Goldstone current
for spontaneously broken supertranslations. The current algebra has an
off-diagonal level structure involving the gravitational cusp anomalous
dimension and the logarithm of the IR cutoff. It is further shown that the
gravitational memory effect is contained as an IR safe observable within the
soft $\mathcal{S}$-matrix.
