We prove that an infinite irreducible Coxeter group cannot be a non-trivial direct product. Let W be a Coxeter group, and write W = W1 × · · · ×Wp ×Wp+1, where W1, . . . ,Wp are infinite irreducible Coxeter groups, and Wp+1 is a finite one. As an application of the main result, we obtain that W1, . . . ,Wp are unique and Wp+1 is unique up to isomorphism. That is, if W = W̃1 × · · · × W̃q × W̃q+1 i...