  We show that the complex hyperbolic metrics defined by Deligne-Mostow and
Thurston on ${\mathcal{M}}_{0,n}$ are singular K\"ahler-Einstein metrics when
${\mathcal{M}}_{0,n}$ is embedded in the Deligne-Mumford-Knudsen
compactification $\overline{\mathcal{M}}_{0,n}$. As a consequence, we obtain a
formula computing the volumes of ${\mathcal{M}}_{0,n}$ with respect to these
metrics using intersection of boundary divisors of
$\overline{\mathcal{M}}_{0,n}$. In the case of rational weights, following an
idea of Y. Kawamata, we show that these metrics actually represent the first
Chern class of some line bundles on $\overline{\mathcal{M}}_{0,n}$, from which
other formulas computing the same volumes are derived.
