  We present a very simple and explicit procedure for nonlocalizing the action
of any theory which can be formulated perturbatively. When the resulting
nonlocal field theory is quantized using the functional formalism --- with unit
measure factor --- its Green's functions are finite to all orders. The
construction also ensures perturbative unitarity to all orders for scalars with
nonderivative interactions, however, decoupling is lost at one loop when vector
and tensor quanta are present. Decoupling can be restored (again, to all
orders) if a suitable measure factor exists. We compute the required measure
factor for pure Yang-Mills at order $g^2$ and then use it to evaluate the
vacuum polarization at one loop. A peculiar feature of our regularization
scheme is that the on-shell tree amplitudes are completely unaffected. This
implies that the nonlocal field theory can be viewed as a highly noncanonical
quantization of the original, local field equations.
