This is a continuation of part I of this paper [see D. Kirby and D. Rees in: Commutative algebra: Syzygies, multiplicities, and birational algebra: AMS-IMS-SIAM Summer Res. Conf. 1992, Contemp. Math. 159, 209-267 (1994; Zbl 0805.13001)]. Let \(M\) be a module of finite length over the local ring \((A,{\mathfrak m}_A)\). The authors give a systematic account to the whole series of Buchsbaum-Rim multiplicities \(e_q(M,E)\) where \(E\) is an arbitrary \(A\)-module. In analogy with the notion of integral closure of \({\mathfrak m}\)-primary ideals they introduce a \(q\)-th integral radical \(\rho_q(M)\subseteq M\) and show that \(e_q(M/N,A)= e_q(M,A)\) iff \(N\subseteq \rho_q(M)\). For instance, \(\rho(M):= \rho_0(M)\) is defined as follows. Let \(0\to L\to F\to M\to 0\) be an exact sequence where \(F\cong A^n\) for some \(n\). The symmetric algebra \({\mathbf S}M\) is the quotient of \({\mathbf S}F\) with respect to the ideal, say, \(J\) generated by \(L\). Then \(\rho(M)\) is the image of \(F\cap J^*\) in \(M\) where \(J^*\) is the integral closure of \(J\). Finally, the degree function associated to the generalized Buchsbaum-Rim multiplicity is studied.