  Spin liquids are exotic phases of quantum matter challenging Landau's
paradigm of symmetry-breaking phase transitions. Despite strong exchange
interactions, spins do not order or freeze down to zero temperature. While
well-established for 1D quantum antiferromagnets, in higher dimension where
quantum fluctuations are less acute, realizing and understanding such states
represent major issues, both theoretical and experimental. In this respect the
simplest nearest-neighbor Heisenberg antiferromagnet Hamiltonian on the highly
frustrated kagome lattice has proven to be a fascinating and inspiring model.
The exact nature of its ground state remains elusive and the existence of a
spin-gap is the first key-issue to be addressed to discriminate between the
various classes of proposed spin liquids. Here, through low-temperature Nuclear
Magnetic Resonance (NMR) contrast experiments on high quality single crystals,
we single out the kagome susceptibility and the corresponding dynamics in the
kagome archetype, the mineral herbertsmithite, ZnCu$_3$(OH)$_6$Cl$_2$. We
firmly conclude that this material does not harbor any spin-gap, which restores
a convergence with recent numerical results promoting a gapless Dirac spin
liquid as the ground state of the Heisenberg kagome antiferromagnet.
