  An E-ring is a unital ring R such that every endomorphism of the underlying
abelian group R^+ is multiplication by some ring-element. The existence of
almost-free E-rings of cardinality greater than 2^{aleph_0} is undecidable in
ZFC. While they exist in Goedel's universe, they do not exist in other models
of set theory. For a regular cardinal aleph_1 <= lambda <= 2^{aleph_0} we
construct E-rings of cardinality lambda in ZFC which have aleph_1-free additive
structure. For lambda = aleph_1 we therefore obtain the existence of
almost-free E-rings of cardinality aleph_1 in ZFC.
