  We completely classify diffeomorphism covariant local nets of von Neumann
algebras on the circle with central charge c less than 1. The irreducible ones
are in bijective correspondence with the pairs of A-D_{2n}-E_{6,8} Dynkin
diagrams such that the difference of their Coxeter numbers is equal to 1. We
first identify the nets generated by irreducible representations of the
Virasoro algebra for c<1 with certain coset nets. Then, by using the
classification of modular invariants for the minimal models by
Cappelli-Itzykson-Zuber and the method of alpha-induction in subfactor theory,
we classify all local irreducible extensions of the Virasoro nets for c<1 and
infer our main classification result. As an application, we identify in our
classification list certain concrete coset nets studied in the literature.
