  The rigidity of the positive mass theorem states that the only complete
asymptotically flat manifold of nonnegative scalar curvature and zero mass is
Euclidean space. We prove a corresponding stability theorem for spaces that can
be realized as graphical hypersurfaces in $\mathbb{R}^{n+1}$. Specifically, for
an asymptotically flat graphical hypersurface $M^n\subset \mathbb{R}^{n+1}$ of
nonnegative scalar curvature (satisfying certain technical conditions), there
is a horizontal hyperplane $\Pi\subset \mathbb{R}^{n+1}$ such that the flat
distance between $M$ and $\Pi$ in any ball of radius $\rho$ can be bounded
purely in terms of $n$, $\rho$, and the mass of $M$. In particular, this means
that if the masses of a sequence of such graphs approach zero, then the
sequence weakly converges (in the sense of currents, after a suitable vertical
normalization) to a flat plane in $\mathbb{R}^{n+1}$. This result generalizes
some of the earlier findings of the second author and C. Sormani and provides
some evidence for a conjecture stated there.
