  This paper is concerned with the study of the random variable $K_n$ denoting
the number of distinct elements in a random sample $(X_1, \dots, X_n)$ of
exchangeable random variables driven by the two parameter Poisson-Dirichlet
distribution, $PD(\alpha,\theta)$. For $\alpha\in(0,1)$, Theorem 3.8 in
\cite{Pit(06)} shows that
$\frac{K_n}{n^{\alpha}}\stackrel{\text{a.s.}}{\longrightarrow}
S_{\alpha,\theta}$ as $n\rightarrow+\infty$. Here, $S_{\alpha,\theta}$ is a
random variable distributed according to the so-called scaled Mittag-Leffler
distribution. Our main result states that $$ \sup_{x \geq 0} \Big|
\ppsf\Big[\frac{K_n}{n^{\alpha}} \leq x \Big] - \ppsf[S_{\alpha,\theta} \leq x]
\Big| \leq \frac{C(\alpha, \theta)}{n^{\alpha}} $$ holds with an explicit
constant $C(\alpha, \theta)$. The key ingredients of the proof are a novel
probabilistic representation of $K_n$ as compound distribution and new, refined
versions of certain quantitative bounds for the Poisson approximation and the
compound Poisson distribution.
