In 1921 G. I. Taylor introduced (with little discussion) the notion that the dispersion of a conserved passive scalar in a turbulent flow is determined by the motion of fluid particles (independent of the molecular diffusivity). Here, a hypothesis of diffusivity independence is introduced, which provides a sufficient condition for the validity of Taylor's approach. The hypothesis, which is supported by DNS data, is that, at high Reynolds number, the mean of the scalar conditional on the velocity is independent of the molecular diffusivity. From this hypothesis it is shown that (at high Reynolds number) the conditional Laplacian of the scalar is zero. This new result has several significant implications for models of turbulent mixing, and for the scalar flux. Primarily, a model of turbulent scalar mixing that is independent of velocity is inconsistent with the hypothesis, and gives rise to a spurious source or (more likely) sink of the scalar flux.