  The dimensionful nature of the coupling in the Einstein-Hilbert action in
four dimensions implies that the theory is non-renormalizable; explicit
calculation shows that beginning at two loop order, divergences arise that
cannot be removed by renormalization without introducing new terms in the
classical action. It has been shown that, by use of a Lagrange multiplier field
to ensure that the classical equation of motion is satisfied in the path
integral, radiative effects can be restricted to one loop order. We show that
by use of such Lagrange multiplier fields, the Einstein-Hilbert action can be
quantized without the occurrence of non-renormalizable divergences. We then
apply this mechanism to a model in which there is in addition to the
Einstein-Hilbert action, a fully covariant action for a self-interacting scalar
field coupled to the metric. It proves possible to restrict loop diagrams
involving internal lines involving the metric to one-loop order; diagrams in
which the scalar field propagates occur at arbitrary high order in the loop
expansion. This model also can be shown to be renormalizable. Incorporating
spinor and vector fields in the same way as scalar fields is feasible, and so a
fully covariant Standard Model with a dynamical metric field can also be shown
to be renormalizable
