  Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by
means of a Riemannian approximation scheme, as limit of Riemannian minimal
surfaces. We study the regularity of Lipschitz, non-characteristic minimal
surfaces which arise as such limits. Our main results are a-priori estimates on
the solutions of the approximating Riemannian PDE and the ensuing $C^{\infty}$
regularity of the sub-Riemannian minimal surface along its Legendrian
foliation.
