  In this paper, we study the Bergman metric of a finite ball quotient
$\mathbb{B}^n/\Gamma$, where $\Gamma \subseteq \mathrm{Aut}(\mathbb{B}^n)$ is a
finite, fixed point free, abelian group. We prove that this metric is
K\"ahler--Einstein if and only if $\Gamma$ is trivial, i.e., when the ball
quotient $\mathbb{B}^n/\Gamma$ is the unit ball $\mathbb{B}^n$ itself. As a
consequence, we establish a characterization of the unit ball among normal
Stein spaces with isolated singularities and abelian fundamental groups in
terms of the existence of a Bergman-Einstein metric.
