For any abelian group A, we prove an asymptotic formula for the number of A-extensions K/Q of bounded discriminant such that the associated norm one torus R_{K/Q}^1 G_m satisfies weak approximation. We will discuss how to parametrise abelian extensions in a convenient way so that the problem reduces to the estimation of a certain character sum. We will then estimate this character sum using Siegel--Walfisz and the large sieve. This is joint work with Nick Rome.