The pressure-temperature phase diagram of the organic Mott insulator kappa-(ET)2Cu2(CN)3, a model system of the spin liquid on triangular lattice, has been investigated by 1H NMR and resistivity measurements. The spin-liquid phase is persistent before the Mott transition to the metal or superconducting phase under pressure. At the Mott transition, the spin fluctuations are rapidly suppressed and the Fermi-liquid features are observed in the temperature dependence of the spin-lattice relaxation rate and resistivity. The characteristic curvature of the Mott boundary in the phase diagram highlights a crucial effect of the spin frustration on the Mott transition.