  We apply the theory of finite dimensional weak C^*-Hopf algebras A as
developed by G. B\"ohm, F. Nill and K. Szlach\'anyi to study reducible
inclusion triples of von-Neumann algebras N \subset M \subset (M\cros\A). Here
M is an A-module algebra, N is the fixed point algebra and \M\cros\A is the
crossed product extension. ``Weak'' means that the coproduct \Delta on A is
non-unital, requiring various modifications of the standard definitions for
(co-)actions and crossed products. We show that acting with normalized positive
and nondegenerate left integrals l\in\A gives rise to faithful conditional
expectations E_l: M-->N, where under certain regularity conditions this
correspondence is one-to-one. Associated with such left integrals we construct
``Jones projections'' e_l\in\A obeying the Jones relations as an identity in
M\cros\A. Finally, we prove that N\subset M always has finite index and depth 2
and that the basic Jones construction is given by the ideal M_1:=M e_l M
\subset M\cros\A, where under appropriate conditions M_1 = M\cros\A. In a
subsequent paper we will show that converseley any reducible finite index and
depth-2 Jones tower of von-Neumann factors (with finite dimensional centers)
arises in this way.
