It is possible to remove ultraviolet divergences from Feynman diagrams by first differentiating them with respect to their external momenta, then integrating with respect to the internal momenta, and finally integrating back with respect to the external momenta. The last (indefinite) integrals involve arbitrary integration constants, which have to be fixed by further prescriptions,e.g. by requiring that there should be neither mass nor charge renormalization. The whole procedure is consistent with the requirements of covariance, unitarity, and causality, so that the ultraviolet divergences should be considered as entirely spurious. The results obtained by the present method are identical to those of renormalized quantum field theory, whenever renormalization is possible, but are reached here with less computational labor. Moreover, this method can also be applied, with the same ease, to nonrenormalizable interactions. Finally, the present method displays very clearly the arbitrariness involved at each stage of the perturbation procedure, which is quite parallel to the semi-phenomenological approach of dispersion theory.