  The aim of this paper is to give an upper bound for the intrinsic diameter of
a surface with boundary immersed in a conformally flat three dimensional
Riemannian manifold in terms of the integral of the mean curvature and of the
length of its boundary. Of particular interest is the application of the
inequality to minimal surfaces in the three-sphere and in the hyperbolic space.
Here the result implies an a priori estimate for connected solutions of
Plateau's problem, as well as a necessary condition on the boundary data for
the existence of such solutions. The proof follows a construction of Miura and
uses a diameter bound for closed surfaces obtained by Topping and Wu-Zheng.
