  We formulate and prove a version of the Segal Conjecture for infinite groups.
For finite groups it reduces to the original version. The condition that G is
finite is replaced in our setting by the assumption that there exists a finite
model for the classifying space underline{E}G for proper actions. This
assumption is satisfied for instance for word hyperbolic groups or cocompact
discrete subgroups of Lie groups with finitely many path components. As a
consequence we get for such groups G that the zero-th stable cohomotopy of the
classifying space BG is isomorphic to the I-adic completion of the ring given
by the zero-th equivariant stable cohomotopy of underline{E}G for I the
augmentation ideal.
