  In this paper we prove that the Gorenstein cyclic quotient singularities of
type \frac 1l (1,..., 1,l-(r-1)) with $l\geq r\geq 2$, have a
\textit{unique}torus-equivariant projective, crepant, partial resolution, which
is ``full'' iff either $l\equiv 0$ mod $% (r-1) $ or $l\equiv 1$ mod $(r-1) $.
As it turns out, if one of these two conditions is fulfilled, then the
exceptional locus of the full desingularization consists of $\lfloor
\frac{l}{r-1} \rfloor$ prime divisors, $\lfloor \frac{l}{r-1}\rfloor - 1$ of
which are isomorphic to the total spaces of $\Bbb{P}_{\Bbb{C}}^1$-bundles over
$\Bbb{P}_{\Bbb{C}%}^{r-2}$. Moreover, it is shown that intersection numbers are
computable explicitly and that the resolution morphism can be viewed as a
composite of successive (normalized) blow-ups. Obviously, the monoparametrized
singularity-series of the above type contains (as its ``first member'') the
well-known Gorenstein singularity defined by the origin of the affine cone
which lies over the $r$-tuple Veronese embedding of $\Bbb{P}_{\Bbb{C}}^{r-1}$.
