  We prove that a positive-definite measure in $\mathbb{R}^n$ with uniformly
discrete support and discrete closed spectrum, is representable as a finite
linear combination of Dirac combs, translated and modulated. This extends our
recent results where we proved this under the assumption that also the spectrum
is uniformly discrete. As an application we obtain that Hof's quasicrystals
with uniformly discrete diffraction spectra must have a periodic diffraction
structure.
