  We study the rotational predissociation of atom - molecule complexes with
very small binding energy. Such complexes can be produced by Feshbach resonance
association of ultracold molecules with ultracold atoms. Numerical calculations
of the predissociation lifetimes based on the computation of the energy
dependence of the scattering matrix elements become inaccurate when the binding
energy is smaller than the energy width of the predissociating state. We derive
expressions that represent accurately the predissociation lifetimes in terms of
the real and imaginary parts of the scattering length and effective range for
molecules in an excited rotational state. Our results show that the
predissociation lifetimes are the longest when the binding energy is positive,
i.e. when the predissociating state is just above the excited state threshold.
