  This paper is the second in a series where we attempt to give a complete
description of the space of all embedded minimal surfaces of fixed genus in a
fixed (but arbitrary) closed 3-manifold. The key for understanding such
surfaces is to understand the local structure in a ball and in particular the
structure of an embedded minimal disk in a ball in $\RR^3$. We show here that
if the curvature of such a disk becomes large at some point, then it contains
an almost flat multi-valued graph nearby that continues almost all the way to
the boundary.
