  In this paper we use the gradient flow equation introduced in [10] to
construct a Morse complex for the Hamiltonian action $\mathbb A_H$ on a mixed
regularity space of loops in the cotangent bundle $T^*M$ of a closed manifold
$M$. Connections between pairs of critical points are realized as genuine
intersections between unstable and stable manifolds, which (despite being
infinite dimensional objects) turn out to have finite dimensional intersection
with good compactness properties. This follows from the existence of an
additional structure, namely a strongly integrable (0)-essential subbundle,
which behaves nicely under the negative gradient flow of the Hamiltonian action
and which is needed to make comparisons. Transversality is achieved by
generically perturbing the negative gradient vector field $-\nabla \mathbb A_H$
of the Hamiltonian action within a class of pseudo-gradient vector fields
preserving all good compactness properties of $-\nabla \mathbb A_H$. This
follows from an abstract transversality result of independent interest for
vector fields on a Hilbert manifold for which stable and unstable manifolds of
rest points are infinite dimensional. The resulting Morse homology is
independent of the choice of the Hamiltonian (and of all other choices but the
choice of the (0)-essential subbundle, which however only changes the
Morse-complex by a shift of the indices) and is isomorphic to the Floer
homology of $T^*M$ as well as to the singular homology of the free loop space
of $M$.
