  We prove that the volume measure of the Brownian sphere is equal to a
constant multiple of the Hausdorff measure associated with the gauge function
$h(r)=r^4\log\log(1/r)$. This shows in particular that the volume measure of
the Brownian sphere is determined by its metric structure. As a key ingredient
of our proofs, we derive precise estimates on moments of the volume of balls in
the Brownian sphere.
