  We describe an explicit UV regularization of the brane singularities for all
4D flat configurations of 6D gauged chiral supergravity compactified on axially
symmetric internal spaces (for which the general solutions are known). All such
solutions have two or fewer co-dimension two singularities, which we resolve in
terms of microscopic co-dimension one cylindrical 4-branes, whose interiors are
capped using the most general possible 4D flat solution of the 6D field
equations. By so doing we show that such a cap is always possible for any given
bulk geometry, and obtain an explicit relationship between the properties of
the capped 4-branes and the various parameters which describe the bulk
solution. We show how these branes generically stabilize the size of the extra
dimensions by breaking the scale invariance which relates classical solutions
to 6D supergravity, and we compute the scalar potential for this modulus in the
4D effective theory. The lifting of this marginal direction provides a natural
realization of the Goldberger-Wise stabilization mechanism in six dimensions.
