Let M be a an open manifold without boundary acted upon by a compact connected Lie group G and let D be a G-equivariant Dirac-type operator on M. Let v : M → LieG be a G-equivariant map such that the corresponding vector field on M does not vanish outside of a compact subset. It was noted by E. Paradan that such a data define an element of compactly supported transversal K-theory. Hence, one can define a topological index by embedding of M into a compact manifold. We introduce a deformed Dirac operator on M, whose index is defined as a distribution on G. Our index theorem states that this analytical index equals the topological index introduced by Paradan. As a main step in the proof we show that the analytical index is invariant under non-compact cobordism of the type considered by Guillemin, Ginzburg and Karshon.