  We consider smooth null cones in a vacuum spacetime that extend to future
null infinity. For such cones that are perturbations of shear-free outgoing
null cones in Schwarzschild spacetimes, we prove bounds for the Bondi energy,
momentum, and rate of energy loss. The bounds depend on the closeness between
the given cone and a corresponding cone in a Schwarzschild spacetime, measured
purely in terms of the differences between certain weighted $L^2$-norms of the
space-time curvature on the cones, and of the geometries of the spheres from
which they emanate. A key step in this paper is the construction of a family of
asymptotically round cuts of our cone, relative to which the Bondi energy is
measured.
