  A consequence of de Finetti's representation theorem is that for every
infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there
exists a probability measure $\mu$ on the Borel sets of $[0,1]$ such that $\bar
X_n = n^{-1} \sum_{i=1}^n X_i$ converges weakly to $\mu$. For a wide class of
probability measures $\mu$ having smooth density on $(0,1)$, we give bounds of
order $1/n$ with explicit constants for the Wasserstein distance between the
law of $\bar X_n$ and $\mu$. This extends a recent result {by} Goldstein and
Reinert \cite{goldstein2013stein} regarding the distance between the scaled
number of white balls drawn in a P\'olya-Eggenberger urn and its limiting
distribution. We prove also that, in the most general cases, the distance
between the law of $\bar X_n$ and $\mu$ is bounded below by $1/n$ and above by
$1/\sqrt{n}$ (up to some multiplicative constants). For every $\delta \in
[1/2,1]$, we give an example of an exchangeable sequence such that this
distance is of order $1/n^\delta$.
