  We study the low-energy spin fluctuations and superfluid density of a series
of pure and Zn-substituted high-$T_c$ superconductors (HTS) using the muon spin
relaxation and $ac$-susceptibility techniques. At a critical doping state,
$p_c$, we find (i) simultaneous abrupt changes in the magnetic spectrum in the
superconducting ground state and (ii) that the slowing down of spin
fluctuations becomes singular at T=0. These results provide experimental
evidence for a novel quantum transition that separates the superconducting
phase diagram of HTS into two distinct ground states.
