Let \(\{{t \atop s}\}\) for not negative natural numbers \(t, s\) denote the Stirling number of the second kind. In the note under review the authors shows how to compute the Stirling number modulo \(p\) if one knows the \(p\)-adic expansions of \(s\) and \(t\). His formula is somewhat complicated, but similar to those satisfied by binomial coefficients first proved by Lucas. As an application he proves congruences of Clausen-Von Staudt's type for poly-Bernoulli numbers \(B_n^k\) occurring in the power series of the \(k\)th polylogarithm.