Exceptional locus of birational morphism is a divisor.

Question

Let $f: V\to W$ be a proper birational morphism of smooth varieties, in a paper I'm reading the author claims that the exceptional locus of $f$ (i.e. the inverse image of the smallest closed set of $W$ outside of which $f$ is an isomorphism) is an effective Cartier divisor, but I don't know how to prove it. I think that in the case of quasi-projective varieties the result is true using Chow's lemma and blow-up approximation(in the sense of theorem 7.17 Chapter 2 of Hartshorne) but I'm not sure if that suffices for the general case.