Self-avoiding rings at the θ point

We determine exactly the numbers C(N,m) of self-avoiding rings of N steps and m nearest-neighbor pairs, for N up to 28, on the square lattice, as well as the average squared radius of gyration for such rings. From these data we locate the {θ} point and determine its exponents as {ν}=0.58±{}0.01, {φ}=0.90±{}0.02. We also calculate a universal quantity for the rings, which was recently determined for the {θ}' point. Our results are inconsistent with the {θ} and {θ}' points being in the same universality class.