  We show that on an open bounded smooth strongly pseudoconvex subset of
$\CC^{n}$, there exists a K\"ahler-Einstein metric with positive Einstein
constant, such that the metric restricted to the Levi distribution of the
boundary is conformal to the Levi form. To achieve this, we solve an associated
complex Monge-Amp\`ere equation with Dirichlet boundary condition. We also
prove uniqueness under some more assumptions on the open set.
