  The phenomenon of superconvergence is proved for all freely infinitely
divisible distributions. Precisely, suppose that the partial sums of a sequence
of free identically distributed, infinitesimal random variables converge in
distribution to a nondegenerate freely infinitely divisible law. Then the
distribution of the sum becomes Lebesgue absolutely continuous with a
continuous density in finite time, and this density can be approximated by that
of the limit law uniformly, as well as in all $L^{p}$-norms for $p>1$, on the
real line except possibly in the neighborhood of one point. Applications
include the global superconvergence to freely stable laws and that to free
compound Poisson laws over the whole real line.
