A general construction, for complete embedded minimal surfaces of nite total curvature in Euclidean three-space, is carried out. In particular, examples with an arbitrary number of ends are given for the rst time. The construction amounts to desingularizing the circles of intersection of a collection of coaxial catenoids and planes. The desingularization process uses Scherk's singly periodic surfaces for an approximate construction which i s subsequently corrected by singular perturbation methods.