  We will study two kinds of actions of a discrete amenable Kac algebra. The
first one is an action whose modular part is normal. We will construct a new
invariant which generalizes a characteristic invariant for a discrete group
action, and we will present a complete classification. The second is a
centrally free action. By constructing a Rohlin tower in an asymptotic
centralizer, we will show that the Connes-Takesaki module is a complete
invariant.
