The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces,operator theory,and spectral theory,and what would be most useful is a characterization in the simplest real-variable terms.We show that the $L^2$ to $L^2$ inequality holds if and only if two $L^2$ to weak-$L^2$ inequalities hold.This is a corollary to a characterization in terms of a two-weight Poisson inequality,and a pair of testing inequalities on bounded functions.Joint work with Eric Sawyer,Chun-Yun Shen,and Ignacio Uriate-Tuero.