The conformal degree of freedom in the gravitational force is a significant object worth studying for the sake of the deeper understanding of quantum physics. After defining the procedure of euclideanization explicitly, we investigate two independent approaches for constructing a well-defined Euclidean path integral formulation, paying special attention to the treatment for the conformal mode. In the first approach, we adopt the policy that the Lorentzian framework is fundamental and that the Euclidean scheme should be formulated so that the latter adjusts to the former. In the second approach, we try to unify both frameworks, leading to a novel path integral formula. It is hoped that both approaches, besides the conformal-rotation technique, serve the Euclidean quantum mechanics and quantum cosmology.