  In a previous paper we exhibited the somewhat surprising property that most
direct links of prime ideals in Gorenstein rings are equimultiple ideals with
reduction number $1$. This led to the construction of large families of
Cohen--Macaulay Rees algebras. The first goal of this paper is to extend this
result to arbitrary Cohen--Macaulay rings. The means of the proof are changed
since one cannot depend so heavily on linkage theory. We then study the
structure of the Rees algebra of these links, more specifically we describe
their canonical module in sufficient detail to be able to characterize
self--linked prime ideals. In the last section multiplicity estimates for
classes of such ideals are established.
