  Neutron scattering from high-quality YBCO6.334 single crystals with a T$_c$
of 8.4 K shows that there is no coexistence with long-range antiferromagnetic
order at this very low, near-critical doping of $\sim$0.055, in contrast to
claims based on local probe techniques. We find that the neutron resonance seen
in optimally doped YBCO7 and underdoped YBCO6.5, has undergone large softening
and damping. It appears that the overdamped resonance, with a relaxation rate
of 2 meV, is coupled to a zero-energy central mode that grows with cooling and
eventually saturates with no change at or below T$_c$. Although a similar
qualitative behaviour is found for YBCO6.35, our study shows that the central
mode is stronger in YBCO6.334 than YBCO6.35. The system remains subcritical
with short-ranged three dimensional correlations.
