\textit{M. G. Kreǐn} investigated in [Mat. Sb., N. Ser. 20(62), 431--495 (1947; Zbl 0029.14103)] the spectrum of self-adjoint extensions \(\tilde S\) within a gap \(J\) of a densely defined symmetric operator \(S\) with finite deficiency indices. The result was generalized by \textit{J. F. Brasche, H. Neidhardt} and \textit{J. Weidmann} in [Math. Z. 214, No. 2, 343--355 (1993; Zbl 0791.47005)] to the case of infinite deficiency indices and pure point spectrum. Regarding the question whether we can put other kinds of spectra into \(J\) (for instance, absolutely continuous or singular continuous spectrum), a positive answer was given by \textit{S. Albeverio, J. F. Brasche} and \textit{H. Neidhardt} in [J. Funct. Anal. 154, No.~1, 130--173 (1998; Zbl 0997.47005)] for the class of (weakly) significant deficient symmetric operators. These studies are essentially completed and extended in the paper under review: the authors present a complete solution of the inverse spectral problem for symmetric operators with several gaps and monotone Weyl function. More exactly, they prove that if \(S\) is the orthogonal sum of a family of (infinitely many) closed symmetric operators unitarily equivalent to a closed symmetric operator \(A\) with equal positive deficiency indices (this special structure of \(S\) has been considered before by \textit{J. F. Brasche, M. M. Malamud} and \textit{H. Neidhardt} in [CMS Conf. Proc. 29, 75--84 (2000; Zbl 0988.47003)]) and there exists a boundary triple \(\{{\mathcal H}_0,\Gamma_0^0,\Gamma_1^0\}\) for \(A^*\) such that the corresponding Weyl function is monotone with respect to the open set \(J\) contained in the resolvent set of \(A^*| _{\ker(\Gamma_0^0)}\), then, for any auxiliary self-adjoint operator \(R\) in some separable Hilbert space \(\eufm R\), \(S\) admits a self-adjoint extension \(\tilde S\) on \(\eufm K\) such that the spectral parts \(\tilde{S}_J:=\tilde{S}| _{E_{\tilde S}(J){\eufm K}}\) and \(R_J:=R| _{E_{R}(J){\eufm R}}\) are unitarily equivalent (\(E_{\tilde S}(\cdot)\) and \(E_{R}(\cdot)\) denote the spectral measures of \(\tilde S\) and \(R\), respectively). The approach is based on an explicit construction of the self-adjoint extension \(\tilde S\) which allows to obtain consistent information on the spectrum of the extension outside the gaps.