H-1 NMR and static susceptibility measurements have been performed in an organic Mott insulator with a nearly isotropic triangular lattice, kappa-(BEDT-TTF)(2)Cu-2(CN)(3), which is a model system of frustrated quantum spins. The static susceptibility is described by the spin S = 1/2 antiferromagnetic triangular-lattice Heisenberg model with the exchange constant J similar to 250 K. Regardless of the large magnetic interactions, the H-1 NMR spectra show no indication of long-range magnetic ordering down to 32 mK, which is 4 orders of magnitude smaller than J. These results suggest that a quantum spin liquid state is realized in the close proximity of the superconducting state appearing under pressure.