Roberto Longo
\def\A{{\cal A}} \def\B{{\cal B}} \def\M{{\cal M}} \def\N{{\cal N}} \def\R{{\cal R}} \def\L{{\cal L}}
Given an irreducible local conformal net $\A$ of von Neumann algebras on $S^1$ and a finite-index conformal subnet $\B\subset\A$, we show that $\A$ is completely rational iff $\B$ is completely rational. In particular this extends a result of F.~Xu for the orbifold construction. By applying previous results of Xu, many coset models turn out to be completely rational and the structure results in \cite{KLM} hold. Our proofs are based on an analysis of the net inclusion $\B\subset\A$; among other things we show that, for a fixed interval $I$, every von Neumann algebra $\R$ intermediate between $\B(I)$ and $\A(I)$ comes from an intermediate conformal net $\L$ between $\B$ and $\A$ with $\L(I)=\R$. We make use of Watatani's result, extended for the purpose from the type $II$ case to arbitrary factors, on the finiteness of the set $\mathfrak I(\N,\M)$ of intermediate subfactors in an irreducible inclusion of factors $\N\subset\M$ with finite Jones index $[\M:\N]$. We provide an explicit bound for the cardinality of $\mathfrak I(\N,\M)$ which depends only on $[\M:\N]$.