  The area of a cross-sectional cut $\Sigma$ of future null infinity
($\mathcal{I}^+$) is infinite. We define a finite, renormalized area by
subtracting the area of the same cut in any one of the infinite number of
BMS-degenerate classical vacua. The renormalized area acquires an anomalous
dependence on the choice of vacuum. We relate it to the modular energy,
including a soft graviton contribution, of the region of $\mathcal{I}^+$ to the
future of $\Sigma$. Under supertranslations, the renormalized area shifts by
the supertranslation charge of $\Sigma$. In quantum gravity, we conjecture a
bound relating the renormalized area to the entanglement entropy across
$\Sigma$ of the outgoing quantum state on $\mathcal{I}^+$.
