  We extract the $Q$-$\bar Q$ potential by using the thermodynamic quantities
obtained in lattice gauge calculations. The potential is tested and found to
give dissociation temperatures that agree well with those from lattice gauge
spectral function analysis. Using such a $Q$-$\bar Q$ potential, we examine the
quarkonium states in a quark-gluon plasma and determine the `quark drip lines'
which separate the region of bound color-singlet $Q\bar Q$ states from the
unbound region. The characteristics of the quark drip lines severely limit the
region of possible bound $Q\bar Q$ states with light quarks to temperatures
close to the phase transition temperature. Bound quarkonia with light quarks
may exist very near the phase transition temperature if their effective quark
mass is of the order of 300-400 MeV and higher.
