  We consider the Einstein-dust equations with positive cosmological constant
$\lambda$ on manifolds with time slices diffeomorphic to an orientable, compact
3-manifold $S$. It is shown that the set of standard Cauchy data for the
Einstein-$\lambda$-dust equations on $S$ contains an open (in terms of suitable
Sobolev norms) subset of data that develop into solutions which admit at future
time-like infinity a space-like conformal boundary ${\cal J}^+$ that is
$C^{\infty}$ if the data are of class $C^{\infty}$ and of correspondingly lower
smoothness otherwise. As a particular case follows a strong stability result
for FLRW solutions. The solutions can conveniently be characterized in terms of
their asymptotic end data induced on ${\cal J}^+$, only a linear equation must
be solved to construct such data. In the case where the energy density
$\hat{\rho}$ is everywhere positive such data can be constructed without
solving any differential equation at all.
