We study charmonium using the standard relativistic formalism in the quenched approximation, on a set of lattices with isotropic lattice spacings ranging from 0.1 to 0.04 fm. We concentrate on the calculation of the hyperfine splitting between ηc and J/ψ, aiming for a controlled continuum extrapolation of this quantity. The splitting extracted from the non-perturbatively improved clover Dirac operator shows very little dependence on the lattice spacing for a ≤ 0.1 fm. The dependence is much stronger for Wilson and tree-level improved clover operators, but they still yield consistent extrapolations if sufficiently fine lattices, a ≤ 0.07 fm (aM(ηc) ≤ 1), are used. Our result for the hyperfine splitting is 77(2)(6) MeV (where Sommer’s parameter, r0 = 0.5 fm, is used to fix the scale). This value remains about 30% below experiment. Dynamical fermions and OZI-forbidden diagrams both contribute to the remainder. Results for the ηc and J/ψ wave functions are also presented.