  Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG)
was launched which replaces the infinite number of Hamiltonian constraints by a
single Master constraint. The MCP is designed to overcome the complications
associated with the non -- Lie -- algebra structure of the Dirac algebra of
Hamiltonian constraints and was successfully tested in various field theory
models. For the case of 3+1 gravity itself, so far only a positive quadratic
form for the Master Constraint Operator was derived. In this paper we close
this gap and prove that the quadratic form is closable and thus stems from a
unique self -- adjoint Master Constraint Operator. The proof rests on a simple
feature of the general pattern according to which Hamiltonian constraints in
LQG are constructed and thus extends to arbitrary matter coupling and holds for
any metric signature. With this result the existence of a physical Hilbert
space for LQG is established by standard spectral analysis.
