  We prove that the free additive convolution of two Borel probability measures
supported on the real line can have a component that is singular continuous
with respect to the Lebesgue measure on the real line only if one of the two
measures is a point mass. The density of the absolutely continuous part with
respect to the Lebesgue measure is shown to be analytic wherever positive and
finite. The atoms of the free additive convolution of Borel probability
measures on the real line have been described by Bercovici and Voiculescu in a
previous paper.
