The renormalization of the vacuum expectation value of the stress-energy tensor of a scalar field propagating in a curved space-time with an arbitrary metric was discussed in a previous paper. A new regularization scheme was introduced which employs a continuation in the dimensionality of space-time implemented with a proper-time representation of the Green's function. Here we present a more general formulation of this method which clarifies its basic features and which explicitly displays the stress tensor as the metric functional derivative of the one-loop action functional. We apply this more general formulation to both the scalar field theory and to the electrodynamic, Maxwell theory. Although the trace of the stress tensor formally vanishes both for the massless scalar field and for the Maxwell field, the trace of the renormalized vacuum expectation value of the stress tensor does not vanish for either theory. These finite-trace anomalies cannot be removed by adding a finite local counterterm into the Lagrange function. The anomalies are intimately related to the infinite scalar counterterms that are needed to render the action finite.