  We discuss the Next-to-Minimal Supersymmetric Standard Model (NMSSM) with a
Peccei-Quinn (PQ) U(1) symmetry. When this symmetry is dynamically broken by
the Higgs mechanism, the resulting pseudo-Nambu-Goldstone boson takes the role
of an axion. Although much of the allowed parameter space for low values of the
PQ scale has been ruled out, many scenarios with a PQ scale > 10^9 GeV remain
untested, allowing the NMSSM PQ axion to provide a solution to the strong CP
problem and be a good dark matter candidate. Unfortunately the new particle
states are so decoupled that they would not be observable at future colliders,
and the NMSSM would appear indistinguishable from the minimal model. However,
we show that in order to maintain vacuum stability, such a model requires that
the heavy Higgs boson states have masses that lie close to approximately mu tan
beta. Therefore, a measurement of the Heavy Higgs boson masses at the LHC would
allow one to either rule out the NMSSM PQ axion, or provide tantalizing
circumstantial evidence for its existence.
