Let K = Q(root-7) and O the ring of integers in K. The prime 2 splits in K, say 2O = p center dot p*. Let A be an elliptic curve defined over K with complex multiplication by O. Assume that A has good ordinary reduction at both p and p*. Write K-infinity for the field generated by the 2(infinity)-division points of A over K and let G = Gal(K-infinity/K). In this paper, by adopting a congruence formula of Yager and De Shalit, we construct the two-variable 2-adic L-function on G. Then by generalizing De Shalit's local structure theorem to the two-variable setting, we prove a two-variable elliptic analogue of Iwasawa's theorem on cyclotomic fields. As an application, we prove that every branch of the two-variable measure has Iwasawa mu invariant zero.