Let (X t ) t > = 0 be a non-singular (not necessarily recurrent) diffusion on R starting at zero, and let ν be a probability measure on R. Necessary and sufficient conditions are established for ν to admit the existence of a stopping time τ * of (X t ) solving the Skorokhod embedding problem, i.e. X τ * has the law ν. Furthermore, an explicit construction of τ * is carried out which reduces to the Azema-Yor construction (Seminaire de Probabilites XIII, Lecture Notes in Mathematics, Vol. 721, Springer, Berlin, p. 90) when the process is a recurrent diffusion. In addition, this τ * is characterized uniquely to be a pointwise smallest possible embedding that stochastically maximizes (minimizes) the maximum (minimum) process of (X t ) up to the time of stopping.