  We prove that a surface of prescribed mean curvature ($H$-surface) with free
boundary on a two-dimensional $C^2$-manifold belongs to $C^{1,\frac{1}{2}}$ up
to that the boundary, provided it is a-priori continuous. We allow the
$H$-surface to meet the manifold non-perpendicularly and the manifold itself to
have a boundary. Our result is optimal according to an example of Hildebrandt
and Nitsche.
