Let \({\mathcal F}\) denote the convolution semigroup of probability distributions on the real line. We prove that no element of \({\mathcal F}\) is prime in the sense that given an \(F\in {\mathcal F}\) one can always find two distributions G,H\(\in {\mathcal F}\) such that F is a convolution factor of G*H but neither of G nor of H. In contrast, \({\mathcal F}\) is known to possess many irreducible elements.