Let $\mu,\nu\in\mathcal P(\mathbb R^n)$ have finite second moments, and suppose $\mu$ is absolutely continuous with respect to $\mathcal L^n$. For the quadratic cost $c(x,y)=|x-y|^2$, there is a proper convex function $\phi:\mathbb R^n\to\mathbb R\cup\{\infty\}$ such that $\nabla\phi$ exists $\mu$-a.e., $(\nabla\phi)_\#\mu=\nu$, and $\nabla\phi$ solves the Monge problem among admissible transport maps.