  We report an inelastic-neutron-scattering and muon-spin-relaxation study of
the effect of 2% spinless (Zn) impurities on the magnetic order and dynamics of
YBa$_2$Cu$_3$O$_{6.6}$, an underdoped high-temperature superconductor that
exhibits a prominent spin-pseudogap in its normal state. Zn substitution
induces static magnetic order at low temperatures and triggers a large-scale
spectral-weight redistribution from the magnetic resonant mode at 38 meV into
uniaxial, incommensurate spin excitations with energies well below the
spin-pseudogap. These observations indicate a competition between
incommensurate magnetic order and superconductivity close to a quantum critical
point. Comparison to prior data on La$_{2-x}$Sr$_x$CuO$_{4}$ suggests that this
behavior is universal for the layered copper oxides and analogous to
impurity-induced magnetic order in one-dimensional quantum magnets.
