  We prove the existence and uniqueness of the canonical conformally covariant
volume measure on the carpet/gasket of a conformal loop ensemble (CLE$_\kappa$,
$\kappa \in (8/3,8)$) which respects the Markov property for CLE. The starting
point for the construction is the existence of the canonical measure on CLE in
the context of Liouville quantum gravity (LQG) previously constructed by the
first author with Sheffield and Werner. As a warm-up, we construct the natural
parameterization of SLE$_\kappa$ for $\kappa \in (4,8)$ using LQG which serves
to complement earlier work of Benoist on the case $\kappa \in (0,4)$.
