  We provide an exact solution of the Efimov spectrum in ultracold gases within
the standard two-channel model for Feshbach resonances. It is shown that the
finite range in the Feshbach coupling makes the introduction of an adjustable
three-body parameter obsolete. The solution explains the empirical relation
between the scattering length a_- where the first Efimov state appears at the
atom threshold and the van der Waals length l_vdw for open channel dominated
resonances. There is a continuous crossover to the closed channel dominated
limit, where the scale in the energy level diagram as a function of the inverse
scattering length 1/a is set by the intrinsic length r* associated with the
Feshbach coupling. Our results provide a number of predictions for
non-universal ratios between energies and scattering lengths that can be tested
in future experiments.
