Neutron spectroscopy and diffuse neutron scattering on herbertsmithite [ZnCu3(OH)(6)Cl-2], a near-ideal realization of the s=1/2 kagome antiferromagnet, reveal the hallmark property of a quantum spin liquid: instantaneous short-ranged antiferromagnetic correlations in the absence of a time-averaged ordered moment. These dynamic antiferromagnetic correlations are weakly dependent of neutron-energy transfer and temperature, and persist up to 25 meV and 120 K. At low energy transfers a shift of the magnetic scattering to low Q is observed with increasing temperature, providing evidence of gapless spinons. It is argued that these observations provide important evidence in favor of resonating-valence-bond theories of (doped) Mott insulators.