  Geometrical frustration, quantum entanglement and disorder may prevent
long-range order of localized spins with strong exchange interactions,
resulting in a novel state of matter. $\kappa$-(BEDT-TTF)$_2$-Cu$_2$(CN)$_3$ is
considered the best approximation of this elusive quantum-spin-liquid state,
but its ground-state properties remain puzzling. Here we present a
multi-frequency electron-spin resonance study down to millikelvin temperatures,
revealing a rapid drop of the spin susceptibility at $T^*=6\,\mathrm{K}$. This
opening of a spin gap, accompanied by structural modifications, suggests the
enigmatic `$6\,\mathrm{K}$-anomaly' as the transition to a valence-bond-solid
ground state. We identify an impurity contribution that becomes dominant when
the intrinsic spins form singlets. Only probing the electrons directly
manifests the pivotal role of defects for the low-energy properties of
quantum-spin systems without magnetic order.
