  We present the first study of the charmonium spectrum using overlap fermions,
on quenched configurations. Simulations are performed on $16^3 \times 72$
lattices, with Wilson gauge action at $\beta$ = 6.3345. We demonstrate that we
have discretization errors under control at about 5%. We obtain 88(4) MeV for
hyperfine splitting using the $r_0$ scale, and 121(6) MeV using the
($1\bar{P}-1\bar{S}$) scale. This paper raises the possibility that the
discrepancy between the lattice results and the experimental value for
charmonium hyperfine splitting can be resolved using overlap fermions to
simulate the charm quark on lattice.
