  In this paper we study the $C^*$-convex set of unital entanglement breaking
(EB-)maps on matrix algebras. General properties and an abstract
characterization of $C^*$-extreme points are discussed. By establishing a
Radon-Nikodym type theorem for a class of EB-maps we give a complete
description of the $C^*$-extreme points. It is shown that a unital EB-map
$\Phi:M_{d_1}\to M_{d_2}$ is $C^*$-extreme if and only if it has Choi-rank
equal to $d_2$. Finally, as a direct consequence of the Holevo form of EB-maps,
we derive a noncommutative analogue of the Krein-Milman theorem for
$C^*$-convexity of the set of unital EB-maps.
