Let {mu((i))(t)}(t >= 0) (i = 1, 2) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that mu((1))(1) = mu((2))(1). Assume furthermore that {mu((1))(t)}(t >= 0) is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then mu((1))(1) = mu((2))(1) for all t >= 0. We end up with a possible application in mathematical finance.