  We show that the support of an irreducible weight module over the twisted
Heisenberg-Virasoro algebra, which has an infinite dimensional weight space,
coincides with the weight lattice and that all nontrivial weight spaces of such
a module are infinite dimensional. As a corollary, we obtain that every
irreducible weight module over the twisted Heisenberg-Virasoro algebra, having
a nontrivial finite dimensional weight space, is a Harish-Chandra module (and
hence is either an irreducible highest or lowest weight module or an
irreducible module from the intermediate series
