  It is shown that for each $N>0$ and for a wide class of Abelian non-compact
locally compact second countable groups $G$ including all infinite countable
discrete ones and $\Bbb R^{d_1}\times\Bbb Z^{d_2}$ with $d_1,d_2\ge 0$, there
exists a weakly mixing probability preserving $G$-action with a homogeneous
spectrum of multiplicity $N$.
