  We prove that for the uniquely ergodic ${\bf R}^d$ action associated with a
primitive substitution tiling of finite local complexity, every measurable
eigenfunction coincides with a continuous function almost everywhere. Thus,
topological weak-mixing is equivalent to measure-theoretic weak-mixing for such
actions. If the expansion map for the substitution is a pure dilation by
$\theta>1$ and the substitution has a fixed point, then failure of weak-mixing
is equivalent to $\theta$ being a Pisot number.
