For a commutative domain
R
R
with
1
1
, an
R
R
-module
B
B
is called a Baer module if
Ext
R
1
(
B
,
T
)
=
0
\operatorname {Ext} _R^1(B,T) = 0
for all torsion
R
R
-modules
T
T
. The structure of Baer modules over arbitrary domains is investigated, and the problem is reduced to the case of countably generated Baer modules. This requires a general version of the singular compactness theorem. As an application we show that over
h
h
-local Prüfer domains, Baer modules are necessarily projective. In addition, we establish an independence result for a weaker version of Baer modules.