  We study the operator that corresponds to the measurement of volume, in
non-perturbative quantum gravity, and we compute its spectrum. The operator is
constructed in the loop representation, via a regularization procedure; it is
finite, background independent, and diffeomorphism-invariant, and therefore
well defined on the space of diffeomorphism invariant states (knot states). We
find that the spectrum of the volume of any physical region is discrete. A
family of eigenstates are in one to one correspondence with the spin networks,
which were introduced by Penrose in a different context. We compute the
corresponding component of the spectrum, and exhibit the eigenvalues
explicitly. The other eigenstates are related to a generalization of the spin
networks, and their eigenvalues can be computed by diagonalizing finite
dimensional matrices. Furthermore, we show that the eigenstates of the volume
diagonalize also the area operator. We argue that the spectra of volume and
area determined here can be considered as predictions of the
loop-representation formulation of quantum gravity on the outcomes of
(hypothetical) Planck-scale sensitive measurements of the geometry of space.
