This is the first in a series of two papers in which we construct a convolution product for associated homogeneous distributions (AHDs) with support in R. In this article, we show that if fa and gb are AHDs with degrees of homogeneity a - 1 and b - 1, the convolution fa * gb exists as an AHD, provided the resulting degree of homogeneity a + b - 1 is not a natural number. Under this restriction, it is found that the convolution product of AHDs is bilinear, bicontinuous and associative. New convolution products are derived for several basis AHDs such as half-line distributions, associated Riesz distributions and associated generalizations of Heisenberg distributions.