  A {\em string graph} is the intersection graph of a family of continuous arcs
in the plane. The intersection graph of a family of plane convex sets is a
string graph, but not all string graphs can be obtained in this way. We prove
the following structure theorem conjectured by Janson and Uzzell: The vertex
set of {\em almost all} string graphs on $n$ vertices can be partitioned into
{\em five} cliques such that some pair of them is not connected by any edge
($n\rightarrow\infty$). We also show that every graph with the above property
is an intersection graph of plane convex sets. As a corollary, we obtain that
{\em almost all} string graphs on $n$ vertices are intersection graphs of plane
convex sets.
