  We present results for the mass spectrum of $c{\bar c}$ mesons simulated on
anisotropic lattices where the temporal spacing $a_t$ is only half of the
spatial spacing $a_s$. The lattice QCD action is the Wilson gauge action plus
the clover-improved Wilson fermion action. The two clover coefficients on an
anisotropic lattice are estimated using mean links in Landau gauge. The bare
velocity of light $\nu_t$ has been tuned to keep the anisotropic, heavy-quark
Wilson action relativistic. Local meson operators and three box sources are
used in obtaining clear statistics for the lowest lying and first excited
charmonium states of $^1S_0$, $^3S_1$, $^1P_1$, $^3P_0$ and $^3P_1$. The
continuum limit is discussed by extrapolating from quenched simulations at four
lattice spacings in the range 0.1 - 0.3 fm. Results are compared with the
observed values in nature and other lattice approaches. Finite volume effects
and dispersion relations are checked.
