  We demonstrate a non-equilibrium phase transition in a dilute thermal atomic
gas. The phase transition, between states of low and high Rydberg occupancy, is
induced by resonant dipole-dipole interactions between Rydberg atoms. The gas
can be considered as dilute as the atoms are separated by distances much
greater than the wavelength of the optical transitions used to excite them. In
the frequency domain we observe a mean-field shift of the Rydberg state which
results in intrinsic optical bistability above a critical Rydberg number
density. In the time domain we observe critical slowing down where the recovery
time to system perturbations diverges with critical exponent
  $\tau=0.53 \pm 0.10$. The atomic emission spectrum of the phase with high
Rydberg occupancy provides evidence for a superradiant cascade.
