  We have mapped the neutron scattering spin spectrum at low-energies in
YBa2Cu3O6.353 (Tc=18 K) where the doping ~0.06 is near the critical value
(pc=0.055) for superconductivity. No coexistence with long range ordered
antiferromagnetism is found. The spins fluctuate on two energy scales, one a
damped spin response with a ~2 meV relaxation rate and the other a central mode
with a relaxation rate that slows to less than 0.08 meV below Tc. The spectrum
mirrors that of a soft mode driving a central mode. Extremely short correlation
lengths, 42+-5 Angstrom in-plane and 8+-2 Angstrom along the c direction, and
isotropic spin orientations for the central mode indicate that the correlations
are subcritical with respect to any second order transition to Neel order. The
dynamics follows a model where damped spin fluctuations are coupled to the slow
fluctuations of regions with correlations shortened by the hole doping.
