Large quantum fluctuations drive the spins in solids into magnetically disordered phases that are not simply paramagnetic. This class of system includes the valence bond crystals and quantum spin liquids, in which spin singlets - the basic unit of entangled pairs of spins - form solids and liquids, respectively. In both phases, geometrical frustration is expected to play a role. So far, very few candidate quantum-spin-liquid materials have been found, including an organic Mott insulator, -(ET) 2 Cu 2 (CN) 3, which is based on a regular triangular lattice. Here, we report a material, -(ET) 2 B(CN) 4, with different geometry - a highly distorted quasi-one-dimensional triangular lattice. The magnetic susceptibility follows that of the spin-1/2 Heisenberg model on this distorted lattice. The material sustains a magnetically disordered Mott insulating state with enhanced quantum fluctuations over a wide temperature range, and undergoes a transition into a spin-gapped phase at 5 K.