  For $0<p<\infty$, $\Psi:[0,\infty)\to(0,\infty)$ and a finite positive Borel
measure $\mu$ on the unit disc $\mathbb{D}$, the Lebesgue--Zygmund space
$L^p_{\mu,\Psi}$ consists of all measurable functions $f$ such that $\lVert f
\rVert_{L_{\mu, \Psi}^{p}}^p =\int_{\mathbb{D}}|f|^p\Psi(|f|)\,d\mu< \infty$.
For an integrable radial function $\omega$ on $\mathbb{D}$, the corresponding
weighted Bergman-Zygmund space $A_{\omega, \Psi}^{p}$ is the set of all
analytic functions in $L_{\mu, \Psi}^{p}$ with $d\mu=\omega\,dA$.
  The purpose of the paper is to characterize bounded (and compact) embeddings
$A_{\omega,\Psi}^{p}\subset L_{\mu, \Phi}^{q}$, when $0<p\le q<\infty$, the
functions $\Psi$ and $\Phi$ are essential monotonic, and $\Psi,\Phi,\omega$
satisfy certain doubling properties. The tools developed on the way to the main
results are applied to characterize bounded and compact integral operators
acting from $A^p_{\omega,\Psi}$ to $A^q_{\nu,\Phi}$, provided $\nu$ admits the
same doubling property as $\omega$.
