General relativity (GR) has previously been extended to incorporate degenerate metrics using Ashtekar's Hamiltonian formulation of the theory. The authors show that a natural alternative choice for the form of the Hamiltonian constraints leads to a theory which agrees with GR for nondegenerate metrics, but differs in the degenerate sector from Ashtekar's original degenerate extension. The Poisson bracket algebra of the alternative constraints closes in the nondegenerate sector, with structure functions that involve the inverse of the spatial triad. Thus, the algebra does not close in the degenerate sector. They find that it must be supplemented by an infinite number of secondary 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 all of the constraints. These states are therefore exact (formal) solutions to this alternative degenerate extension of quantum gravity, even though they are not solutions to the usual vector constraint.