The kagome Heisenberg antiferromagnet is a leading candidate in the search for a spin system with a quantum spin-liquid ground state. The nature of its ground state remains a matter of great debate. We conducted 17-O single crystal NMR measurements of the S=1/2 kagome lattice in herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$, which is known to exhibit a spinon continuum in the spin excitation spectrum. We demonstrate that the intrinsic local spin susceptibility $chi_{kagome}$ deduced from the 17-O NMR frequency shift asymptotes to zero below temperature T ~ 0.03 J, where J ~ 200 K is the Cu-Cu super-exchange interaction. Combined with the magnetic field dependence of $chi_{kagome}$ we observed at low temperatures, these results imply that the kagome Heisenberg antiferromagnet has a spin-liquid ground state with a finite gap.
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