The statistics of a long closed self-avoiding walk (SAW) or polymer ri
ng on a d-dimensional lattice obeys hyperscaling. The combination p(N)
[R(2)](d/2)(N)mu(-N) (where p(N) is the number of configurations of a
n oriented and rooted N-step ring, [R(2)](N) a typical average size sq
uared, and mu the SAW effective connectivity constant of the lattice)
is equal for N --> infinity to a lattice-dependent constant times a un
iversal amplitude A(d). The latter amplitude is calculated directly fr
om the minimal continuous Edwards model to second order in epsilon = 4
- d. The case of rings at the upper critical dimension d = 4 is also
studied. The results are checked against field-theoretical calculation
s, and former simulations. As a consequence, we show that the universa
l constant lambda appearing to second order in epsilon in all critical
phenomena amplitude ratios is equal to lambda = 1/3 psi'(1/3)-2/9 pi(
2).