We study the ODE/IM correspondence for all states of the quantum gˆ-KdV model, where gˆ is the affinization of a simply-laced simple Lie algebra g. We construct quantum gˆ-KdV opers as an explicit realization of the class of opers introduced by Feigin and Frenkel, which are defined by fixing the singularity structure at 0 and ∞, and by allowing a finite number of additional singular terms with trivial monodromy. We prove that the generalized monodromy data of the quantum gˆ-KdV opers satisfy the Bethe Ansatz equations of the quantum gˆ-KdV model. The trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities.