  We characterize the absolute continuity of convolution products of orbital
measures on the classical, irreducible Riemannian symmetric spaces $G/K$ of
Cartan type $III$, where $G$ is a non-compact, connected Lie group and $K$ is a
compact, connected subgroup. By the orbital measures, we mean the uniform
measures supported on the double cosets, $KzK,$ in $G$. The characterization
can be expressed in terms of dimensions of eigenspaces or combinatorial
properties of the annihilating roots of the elements $z$.
  A consequence of our work is to show that the convolution product of any
rank% $G/K,$ continuous, $K$-bi-invariant measures is absolutely continuous in
any of these symmetric spaces, other than those whose restricted root system is
type $A_{n}$ or $D_{3}$, when rank$G/K$ $+1$ is needed.
