  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.
