We show that any countable subgroup of the multiplicative group
$\mathbb{R}_+^{\times}$ of positive real numbers can be realized as the
fundamental group $\mathcal{F}(A)$ of a separable simple unital $C^*$-algebra
$A$ with unique trace. Furthermore for any fixed countable subgroup $G$ of
$\mathbb{R}_+^{\times}$, there exist uncountably many mutually nonisomorphic
such algebras $A$ with $G = \mathcal{F}(A)$.