  A fundamental connection between superconductivity and quantum spin
fluctuations in underdoped cuprates, is revealed. A variational calculation
shows that {\em Cooper pair hopping} strongly reduces the local magnetization
$m_0$. This effect pertains to recent neutron scattering and muon spin rotation
measurements in which $m_0$ varies weakly with hole doping in the poorly
conducting regime, but drops precipitously above the onset of
superconductivity.
