  This paper proposes a regularization of the Monge-Amp\`ere equation in planar
convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations.
The regularized problem possesses a unique strong solution $u_\varepsilon$ and
is accessible to the discretization with finite elements. This work establishes
locally uniform convergence of $u_\varepsilon$ to the convex Alexandrov
solution $u$ to the Monge-Amp\`ere equation as the regularization parameter
$\varepsilon$ approaches $0$. A mixed finite element method for the
approximation of $u_\varepsilon$ is proposed, and the regularized finite
element scheme is shown to be locally uniformly convergent. Numerical
experiments provide empirical evidence for the efficient approximation of
singular solutions $u$.
