  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.
