Let $\tilde{\h}$ be a Cartan subalgebra of an affine Kac-Moody algebra without containing the center. We classify the module structures on $U(\tilde{\h})$ for all affine Kac-Moody algebras which are free $U(\tilde{\h})$-modules of rank 1. Actaully these modules exist only for type $A_l^{(1)}(l\ge 1)$ or $C_l^{(1)}(l\ge 2)$ affine Kac-Moody algebras. Most of these modules are irreducible non-weight modules. This is a joint work with Yan-An Cai and Kaiming Zhao.