  We derive a static potential for a heavy quark-antiquark pair propagating in
Minkowski time at finite temperature, by defining a suitable gauge-invariant
Green's function and computing it to first non-trivial order in Hard Thermal
Loop resummed perturbation theory. The resulting Debye-screened potential could
be used in models that attempt to describe the ``melting'' of heavy quarkonium
at high temperatures. We show, in particular, that the potential develops an
imaginary part, implying that thermal effects generate a finite width for the
quarkonium peak in the dilepton production rate. For quarkonium with a very
heavy constituent mass M, the width can be ignored for T \lsim g^2 M/12\pi,
where g^2 is the strong gauge coupling; for a physical case like bottomonium,
it could become important at temperatures as low as 250 MeV. Finally, we point
out that the physics related to the finite width originates from the
Landau-damping of low-frequency gauge fields, and could be studied
non-perturbatively by making use of the classical approximation.
