  We prove that the supremum of principal curvatures of a minimal embedded disc
in hyperbolic three-space spanning a quasicircle in the boundary at infinity is
estimated in a sublinear way by the norm of the quasicircle in the sense of
universal Teichm\"uller space, if the quasicircle is sufficiently close to
being the boundary of a totally geodesic plane. As a by-product we prove that
there is a universal constant C independent of the genus such that if the
Teichm\"uller distance between the ends of a quasi-Fuchsian manifold $M$ is at
most C, then $M$ is almost-Fuchsian. The main ingredients of the proofs are
estimates on the convex hull of a minimal surface and Schauder-type estimates
to control principal curvatures.
