  The Bergman projection $P_\alpha$, induced by a standard radial weight, is
bounded and onto from $L^\infty$ to the Bloch space $\mathcal{B}$. However,
$P_\alpha: L^\infty\to \mathcal{B}$ is not a projection. This fact can be
emended via the boundedness of the operator
$P_\alpha:BMO_2(\Delta)\to\mathcal{B}$, where $BMO_2(\Delta)$ is the space of
functions of bounded mean oscillation in the Bergman metric.
  We consider the Bergman projection $P_\omega$ and the space
$BMO_{\omega,p}(\Delta)$ of functions of bounded mean oscillation induced by
$1<p<\infty$ and a radial weight $\omega\in\mathcal{M}$. Here $\mathcal{M}$ is
a wide class of radial weights defined by means of moments of the weight, and
it contains the standard and the exponential-type weights. We describe the
weights such that $P_\omega:BMO_{\omega,p}(\Delta)\to\mathcal{B}$ is bounded.
They coincide with the weights for which $P_\omega: L^\infty \to \mathcal{B}$
is bounded and onto. This result seems to be new even for the standard radial
weights when $p\ne2$.
