  For a given cardinal $\lambda$ and a torsion abelian group $K$ of cardinality
less than $\lambda$, we present, under some mild conditions (for example
$\lambda=\lambda^{\aleph_0}$), boundedly endo-rigid abelian group $G$ of
cardinality $\lambda$ with $Tor(G)=K$. Essentially, we give a complete
characterization of such pairs $(K, \lambda)$. Among other things, we use a
twofold version of the black box. We present an application of the construction
of boundedly endo-rigid abelian groups. Namely, we turn to the existing problem
of co-Hopfian abelian groups of a given size, and present some new classes of
them, mainly in the case of mixed abelian groups. In particular, we give useful
criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and
completely determine cardinals $\lambda> 2^{\aleph_{0}}$ for which there is a
co-Hopfian abelian group of size $\lambda$.
