Summary: Let \(A : = \mathbb{Z} [z_1, \ldots, z_r] \supset \mathbb{Z}\) be a finitely generated integral domain over \(\mathbb{Z}\), let \(K\) denote its quotient field, and \(K^\ast\) the multiplicative group of non-zero elements of \(K\). Let \(\Gamma\) be a finitely generated subgroup of \(K^\ast\), and let \(\overline{\Gamma}\) denote the division group of \(\Gamma\). Let \(F(X, Y) \in A [X, Y]\) be a polynomial. In [Astérisque 24–25, 187–210 (1975; Zbl 0315.14005)] P. Liardet proved that under some natural conditions on \(F\) the equation \[ F(x, y) = 0\quad \text{with}\quad x, y \in \overline{\Gamma} \] has only finitely many solutions. The proof of Liardet was ineffective. In 2009 an effective version of Liardet’s Theorem has been proved by the author et al. [Math. Proc. Camb. Philos. Soc. 147, No. 1, 69–94 (2009; Zbl 1177.11054)] in the case when \(\Gamma \subset \overline{\mathbb Q}\). In the present paper an effective version of Liardet’s theorem is proved in the general case.