This is the second in a series of two papers in which we construct a convolution product for the set H' (R) of associated homogeneous distributions (AHDs) with support in R. In Part I we showed that if f a and gb are AHDs with degrees of homogeneity a - 1 and b - 1, the convolution fa * gb exists as an AHD, if the resulting degree of homogeneity a + b-1 N. In this article, we develop a functional extension process, based on the Hahn-Banach theorem, to give a meaning to the convolution product of two AHDs of degrees a - 1 and b - 1, in the critical case that a + b - 1 N. With respect to this construction, the structure (H'(R), *) is shown to be closed.