  General relativity has previously been extended to incorporate degenerate
metrics using Ashtekar's hamiltonian formulation of the theory. In this letter,
we show that a natural alternative choice for the form of the hamiltonian
constraints leads to a theory which agrees with GR for non-degenerate metrics,
but differs in the degenerate sector from Ashtekar's original degenerate
extension. The Poisson bracket algebra of the alternative constraints closes in
the non-degenerate sector, with structure functions that involve the {\it
inverse} of the spatial triad. Thus, the algebra does {\it not} close in the
degenerate sector. We find that it must be supplemented by an infinite number
ofsecondary constraints, which are shown to be first class (although their
explicit form is not worked out in detail). All of the constraints taken
together are implied by, but do not imply, Ashtekar's original form of
constraints. Thus, the alternative constraints give rise to a different
degenerate extension of GR. In the corresponding quantum theory, the single
loop and intersecting loop holonomy states found in the connection
representation satisfy {\it all} of the constraints. These states are therefore
exact (formal) solutions to this alternative degenerate extension of quantum
gravity, even though they are {\it not} solutions to the usual vector
constraint.
