  We define a physical Hilbert space for the three-dimensional lattice gravity
of Ponzano and Regge and establish its isomorphism to the ones in the $ISO(3)$
Chern-Simons theory. It is shown that, for a handlebody of any genus, a
Hartle-Hawking-type wave-function of the lattice gravity transforms into the
corresponding state in the Chern-Simons theory under this isomorphism. Using
the Heegaard splitting of a three-dimensional manifold, a partition function of
each of these theories is expressed as an inner product of such wave-functions.
Since the isomorphism preserves the inner products, the partition function of
the two theories are the same for any closed orientable manifold. We also
discuss on a class of topology-changing amplitudes in the lattice gravity and
their relation to the ones in the Chern-Simons theory.
