  We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic
$\mathbf{Z}_p$-extension of the CM field, where $p$ is a prime of good,
ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to
even order, the two variable main conjecture of Rubin implies that the
Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension
is a torsion Iwasawa module. When the order of vanishing is odd, work of
Greenberg shows that it is not a torsion module. In this paper we show that in
the case of odd order of vanishing the dual of the Selmer group has rank
exactly one, and we prove a form of the Iwasawa main conjecture for the torsion
submodule.
