  We analyze the low frequency part of charmonium spectral functions on large
lattices close to the continuum limit in the temperature region $1.5\lesssim
T/T_c\lesssim 3$ as well as for $T \simeq 0.75T_c$. We present evidence for the
existence of a transport peak above $T_c$ and its absence below $T_c$. The
heavy quark diffusion constant is then estimated using the Kubo formula. As
part of the calculation we also determine the temperature dependence of the
signature for the charmonium bound state in the spectral function and discuss
the fate of charmonium states in the hot medium.
