  A rigorous algebraic proof of the full finiteness in all orders of
perturbation theory is given for the Yang-Mills-Chern-Simons theory in a
general three-dimensional Riemannian manifold. We show the validity of a trace
identity, playing the role of a local form of the Callan-Symanzik equation, in
all loop orders, which yields the vanishing of the beta-functions associated to
the topological mass and gauge coupling constant as well as the anomalous
dimensions of the fields.
