  Impurities, which are inherently present in any real material, may play an
important role in the magnetism of frustrated spin systems with spin-liquid
ground states. We address the impurity issue in quantum kagome antiferromagnets
by investigating ZnCu$_3$(OH)$_6$SO$_4$ (Zn-brochantite) by means of muon spin
spectroscopy. We show that muons couple to the impurity magnetism, originating
from Cu-Zn intersite disorder, and that the impurities are highly correlated
with the kagome spins, allowing us to probe the intrinsic kagome physics via a
Kondo-like effect. The low-temperature plateau in local susceptibility
identifies the spin-liquid ground state as being gapless. The corresponding
spin fluctuations exhibit an unconventional spectral density and an intriguing
field dependence.
