  The potential between charm and anti-charm quarks is calculated
non-perturbatively using physical, rather than static quarks at temperatures on
both sides of the deconfinement transition $T_{\rm C}$, using a lattice
simulation with 2+1 dynamical quark flavours. We used the HAL QCD
time-dependent method, originally developed for inter-nucleon potentials. Our
lattices are anisotropic, with temporal lattice spacing less than the spatial
one which enhances the information content of our correlators at each
temperature. Local-extended charmonium correlators were calculated efficiently
by contracting propagators in momentum rather than coordinate space. We find no
significant variation in the central potential for temperatures in the confined
phase. As the temperature increases into the deconfinement phase, the potential
flattens, consistent with the expected weakening interaction. We fit the
potential to both the (a) Cornell and (b) Debye-screened potential forms, with
the latter better reproducing the data. The zero temperature string tension
obtained from (a) agrees with results obtained elsewhere, and it decreases with
temperature, but at a slower rate than from the static quark approximation. The
Debye mass from (b) is close to zero for small temperatures, but starts to
increase rapidly around $T_{\rm C}$. The spin-dependent potential is found to
have a repulsive core and a distinct temperature dependence above $T_{\rm C}$
at distances $\sim 1$ fm.
