The paper deals with the Sturm-Liouville operators associated with the general, three-coefficient differential expression \[ \frac{1}{r(x)} \Bigl[ -\frac{d}{dx} p(x) \frac{d}{dx} + q(x)\Bigr]\text{ for a.e. }x\in (a,b) \subseteq \mathbb R, \] where the functions \(\frac{1}{p}\), \(q\), and \(r\) belong to\(L^1_{\mathrm{loc}}((a,b); dx)\), \(p\) and \(r\) are positive, and \(q\) is real-valued. The minimal operator \(T_{\min}\) is assumed in addition to be strictly positive. The authors obtain an explicit description of the Krein-von Neumann extensions of \(T_{\min}\) in terms of generalized boundary values. As examples, a generalized Bessel operator, a singular operator relevant in the context of acoustic black holes, and the Jacobi operator are considered.