  We exhibit a novel solution of the strong CP problem, which does not involve
any massless particles. The low energy effective Lagrangian of our model
involves a discrete spacetime independent axion field which can be thought of
as a parameter labeling a dense set of $\theta$ vacua. In the full theory this
parameter is seen to be dynamical, and the model seeks the state of lowest
energy, which has $\theta_{eff} = 0$. The processes which mediate transitions
between $\theta$ vacua involve heavy degrees of freedom and are very slow.
Consequently, we do not know whether our model can solve the strong CP problem
in a universe which has been cool for only a finite time. We present several
speculations about the cosmological evolution of our model.
