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.
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