It is shown that if H, K are any finitely generated subgroups of a free group F and U is any cyclic subgroup of F, then any intersection Hg U l Kg U of double 1 2 Ž . cosets contains only a finite number of double cosets H l K gU, and an explicit upper bound for this number is given in terms of the ranks of H and K and a generator of U. This result is then applied to the intersection of finitely generated subgroups H, K of a free product with amalgamation G s A ) B with A free and U U maximal cyclic in A. Under the assumption that H and K intersect all conjugates of U trivially, an upper estimate is established for the ''Karrass᎐Solitar rank'' of H l K in terms of the KS-ranks of H and K, a generator of U, and max rank g y1 Hg l A , max rank g y1 Kg l A . Ä 4 Ä 4 Ž . Ž . ggG g gG Here the Karrass᎐Solitar rank of H F A ) B is defined to be the size of a natural U set of generating subgroups of H, afforded by the Karrass᎐Solitar subgroup theorem for amalgamated products A ) B. U 1 2 Ž . any two cosets of H, K respectively, is either empty or a single right coset of H l K. However, in the more general situation of double cosets Hg U, Kg U, where U is a further subgroup of G, the intersection Hg U l 1 2 1 165