The quest for entangled spin excitations has stimulated intense research on frustrated magnetic systems. For almost two decades, the triangular-lattice Mott insulator κ-(BEDT-TTF) Cu (CN) has been one of the hottest candidates for a gapless quantum spin liquid with itinerant spinons. Very recently, however, this scenario was overturned as electron-spin-resonance (ESR) studies unveiled a spin gap, calling for reevaluation of the magnetic ground state. Here we achieve a precise mapping of this spin-gapped phase through the Mott transition by ultrahigh-resolution strain tuning. Our transport experiments reveal a reentrance of charge localization below T  = 6 K associated with a gap size of 30-50 K. The negative slope of the insulator-metal boundary, dT /dp < 0, evidences the low-entropy nature of the spin-singlet ground state. By tuning the enigmatic '6K anomaly' through the phase diagram of κ-(BEDT-TTF) Cu (CN) , we identify it as the transition to a valence-bond-solid phase, in agreement with previous thermal expansion and magnetic resonance studies. This spin-gapped insulating state persists at T → 0 until unconventional superconductivity and metallic transport proliferate.