Let \(s(n, m)\) denote the unsigned Stirling numbers of the first kind. For any \(\eta> 0\) and natural number \(v\), the following asymptotic formula holds uniformly \[ {s(n, m)\over n!}= {1\over n} \sum_{0\leq k\leq v} {\Pi_{m,k} (\log n)\over n^k}+ O \Biggl({(\log n)^m\over m!n^{v+ 2}}\Biggr) \] for \(1\leq m\leq \eta\log n\), where \(\Pi_{m, k}(x)\) are explicitly given polynomials in \(x\) of degree \(m- 1\). Since the asymptotic behaviour of \(\Pi_{m, k}(\log n)\) is unclear for \(m= \Omega(\log n)\), a uniform asymptotic expansion for \(\Pi_{m, 0}(\log n)\) is also given. These results can be interpreted as saying that the Stirling numbers of the first kind are asymptotically Poisson distributed of parameter \(\log n\). The proof uses a variant of the saddle point method.