Moufang sets are split doubly transitive permutation groups, or equivalently, groups with a split BN-pair of rank one. In this paper, we study so-called special Moufang sets with abelian root groups, under the model-theoretic restriction that the groups have finite Morley rank. These groups have a natural base field, and we classify them under the additional assumption that the base field is infinite. The result is that the group is isomorphic to PSL 2 (K) over some algebraically closed field K.