  We establish a regularity result for the metric on any 4-dimensional extremal
K\"ahler manifold, and a weak compactness theorem on the space of such metrics.
Specifically, the sectional curvature at a point is bounded when the quantity
$L^2(|\Riem|)$ in a surrounding ball is sufficiently small compared to the
pointwise norm of its scalar curvature. Consequently sequences of 4-dimensional
extremal K\"ahler metrics with uniformly bounded Calabi energies and scalar
curvature have convergent subsequences in the Gromov-Hausdorff topology.
Gromov-Hausdorff limits are length spaces with the structure of Riemannian
orbifolds away from finitely many point-like singularities of unknown
structure.
