Let \(C[r_ 1,r_ 2]\) denote the real Banach space of continuous functions on \([r_ 1,r_ 2]\) with \(-\infty \leq r_ 1<r_ 2\leq +\infty\), equipped with the supremum norm. Let \(\alpha\) and \(\beta\) be continuous functions on \((r_ 1,r_ 2)\), with \(\alpha\) positive on \((r_ 1,r_ 2)\). The following linear operator \(A: D(A)\subset C[r_ 1,r_ 2]\to C[r_ 1,r_ 2]\) is considered. \(D(A):=\{u\in C[r_ 1,r_ 2];\quad u\in C^ 2(r_ 1,r_ 2)\quad and\quad \lim_{x\to r_ i}\alpha u''(x)+\beta u'(x)=0,\quad i=1,2,\},\) \(Au(x):=\alpha u''(x)+\beta (x)u'(x),\) \(x\in (r_ 1,r_ 2)\), \(u\in D(A)\). The boundary conditions used here are a special case of boundary conditions introduced by A. D. Ventcel in [Teor. Verojatn. Primen 4, 172-185 (1959; Zbl 0089.134)]. In theorem 2, necessary and sufficient conditions on \(\alpha\) and \(\beta\) are given, for A to be the infinitesimal generator of a (positive) \(C_ 0\)-semigroup of contractions on \(C[r_ 1,r_ 2]\). The case \(-\infty <r_ 1<r_ 2<+\infty\) is further investigated and comparison with an earlier result of R. Martini [Differential operators degenerating at the boundary as infinitesimal generators of semigroups, Thesis TH Delf (1975), see also Zbl 0288.47043] is made.