[Part I, cf. the first and the third author, ibid. 36, 767-783 (1983; Zbl 0507.58037).] - Let \(L^ 2\) be the Hilbert space of square- integrable real-valued functions on [0,1]. If \(q\in L^ 2\) and \((a,b)\in {\mathbb{R}}^ 2\) are fixed, then the Sturm-Liouville problem \(- y''+qy=\lambda y\), \(0\leq x\leq 1\), with boundary conditions a y(0)\(+y'(0)=0\), b y(1)\(+y'(1)=0\), has simple eigenvalues \(\nu_ n=\nu_ n(a,b,q)\), \(n\geq 0\), increasing to \(\infty\) with corresponding eigenfunctions \(h_ n(x,a,b,q)\). The authors consider two aspects of the inverse problem: The first objective is to describe the spectral classes with fixed boundary conditions (a,b). These are the manifolds \(M_{a,b}(p)=\{q\in L^ 2:\nu_ n(a,b,q)=\nu (a,b,p),n\geq 0\}\) of all potentials q having the same spectrum as some fixed potential p. The second objective is to describe all possible spectra \(\nu_ n\), \(n\nearrow 0\), for fixed and for variable boundary conditions.