  We present evidence that the antiferromagnetic spin correlations in optimally
doped YBa_2Cu_3O_7 and underdoped YBa_2Cu_4O_8 develop a surprisingly strong
anisotropy in the superconducting state. Comparing the ratio of the nuclear
spin-lattice relaxation rates of the planar copper and oxygen, measured at the
lowest and highest temperatures as well as at T_c, we conclude that the
antiferromagnetic in-plane correlations vanish as the temperature goes to zero.
This observation is corroborated by the measurement of the copper linewidth in
YBa_2Cu_4O_8. In contrast, the out-of-plane correlations do not change
appreciably between T=T_c and T=0. Within a model of fluctuating fields this
extreme anisotropy of the antiferromagnetic correlations also explains the
observed temperature dependence of the anisotropy of the copper relaxation
measured in a low external magnetic field.
