  Confirming a conjecture of Vera T. S\'os in a very strong sense, we give a
complete solution to Tur\'an's hypergraph problem for the Fano plane. That is
we prove for $n\ge 8$ that among all $3$-uniform hypergraphs on $n$ vertices
not containing the Fano plane there is indeed exactly one whose number of edges
is maximal, namely the balanced, complete, bipartite hypergraph. Moreover, for
$n=7$ there is exactly one other extremal configuration with the same number of
edges: the hypergraph arising from a clique of order $7$ by removing all five
edges containing a fixed pair of vertices.
  For sufficiently large values $n$ this was proved earlier by F\"uredi and
Simonovits, and by Keevash and Sudakov, who utilised the stability method.
