Let E be a complex elliptic curve and let exp be its exponential map. The map exp is a homomorphism from the additive group of the complex numbers onto E. Consider a subgroup G of E obtained as the image under exp of a real line through the origin having trivial intersection with the kernel of exp. Then G is dense in E in the Euclidean topology. Consider the structure on E with a predicate for each of the Zariski closed subsets of its cartesian powers and a predicate for G. We show that if E does not have complex multiplication, is invariant under complex conjugation and a version of the Schanuel Conjecture holds for exp, then the theory of this structure is omega-stable and has quantifier elimination after adding predicates for the existentially definable sets. To do this we write axioms for the theory by means of a "predimension function", the proof that these axioms hold in the structure uses facts from the theory of analytic sets and o-minimality. Although conditionally, this provides new examples of stable expansions of the complex field.