  We consider the infrared and ultraviolet behaviour of the effective quantum
field theory of a single $Z_2$ symmetric scalar field. In a previous paper we
proved to all orders in perturbation theory the renormalizability of massive
effective scalar field theory using Wilson's exact renormalization group
equation. Here we show that away from exceptional momenta the massless theory
is similarly renormalizable, and we prove detailed bounds on Green's functions
as arbitrary combinations of exceptional Euclidean momenta are approached. As a
corollary we also prove Weinberg's Theorem for the massive effective theory, in
the form of bounds on Green's functions at Euclidean momenta much greater than
the particle mass but below the naturalness scale of the theory.
