Let {mu((i))(t)}t >= 0 (i = 1.2) be continuous convolution semigroups on a simply connected nilpotent Lie group G. Suppose that mu((1))(1) = mu((2))(1) and that {mu((1))(t)}(t) >= 0 is a Gaussian semigroup (in the sense that its generating distribution just consists of a primitive distribution and a second order differential operator). Then mu((1))(t) = mu((2))(t) for all t >= 0.