We consider the smallest values taken by the Jones index for an inclusion of local conformal nets of von Neumann algebras on $S^1$ and show that these values are quite more restricted than for an arbitrary inclusion of factors. Below 4, the only non-integer admissible value is $4\cos^2 \pi/10$, which is known to be attained by a certain coset model. Then no index value is possible in the interval between 4 and $3 +\sqrt{3}$. The proof of this result based on $\alpha$-induction arguments. In the case of values below 4 we also give a second proof of the result. In the course of the latter proof we classify all possible unitary braiding symmetries on the A D E tensor categories, namely the ones associated with the even vertices of the $A_n$, $D_{2n}$, $E_6$, $E_8$ Dynkin diagrams.