  We construct a lattice Dirac operator of overlap type that describes the
propagation of a Dirac fermion in an external gravitational field. The local
Lorentz symmetry is manifestly realized as a lattice gauge symmetry, while it
is believed that the general coordinate invariance is restored only in the
continuum limit. Our doubler-free Dirac operator satisfies the conventional
Ginsparg-Wilson relation and possesses gamma_5 hermiticity with respect to the
inner product, which is suggested by the general coordinate invariance. The
lattice index theorem in the presence of a gravitational field holds, and the
classical continuum limit of the index density reproduces the Dirac genus.
Reduction to a single Majorana fermion is possible for 8k+2 and 8k+4
dimensions, but not for 8k dimensions, which is consistent with the existence
of the global gravitational/gauge anomalies in 8k dimensions. Other Lorentz
representations, such as the spinor-vector and the bi-spinor representations,
can also be treated. Matter fields with a definite chirality (with respect to
the lattice-modified chiral matrix) are briefly considered.
