  We establish a curvature estimate for classical minimal surfaces with total
boundary curvature less than 4\pi. The main application is a bound on the genus
of these surfaces depending solely on the geometry of the boundary curve. We
also prove that the set of simple closed curves with total curvature less than
4\pi and which do not bound an orientable compact embedded minimal surface of
genus greater than g, for any given g, is open in the C^{2,\alpha} topology.
