ERRATUM: Theorem 4.9 is stated too generally: the place $v$ should be assumed to be of degree $1$, that is to say, the associated completion is in fact isomorphic to the real numbers or the field of p-adic numbers. (The proof invokes a mesaure classification result that applies only with this hypothesis.) The Theorem stated in the introduction (Duke's theorem for cubic fields) is not affected by this.