A quantum spin-liquid phase is an intriguing possibility for a system of strongly interacting magnetic units in which the usual magnetically ordered ground state is avoided owing to strong quantum fluctuations. It was first predicted theoretically for a triangular-lattice model with antiferromagnetically coupled S = 1/2 spins(1). Recently, materials have become available showing persuasive experimental evidence for such a state(2). Although many studies show that the ideal triangular lattice of S = 1/2 Heisenberg spins actually orders magnetically into a three-sublattice, non-collinear 1206 arrangement, quantum fluctuations significantly reduce the size of the ordered moment(3). This residual ordering can be completely suppressed when higher-order ring-exchange magnetic interactions are significant, as found in nearly metallic Mott insulators(4). The layered molecular system kappa-(BEDT-TTF)(2)Cu(2)(CN)(3) is a Mott insulator with an almost isotropic, triangular magnetic lattice of spin-1/2 BEDT-TTF dimers(5) that provides a prime example of a spin liquid formed in this way(6-11). Despite a high-temperature exchange coupling, J, of 250 K (ref. 6), no obvious signature of conventional magnetic ordering is seen down to 20 mK (refs 7, 8). Here we show, using muon spin rotation, that applying a small magnetic field to this system produces a quantum phase transition between the spin-liquid phase and an antiferromagnetic phase with a strongly suppressed moment. This can be described as Bose-Einstein condensation of spin excitations with an extremely small spin gap. At higher fields, a second transition is found that suggests a threshold for deconfinement of the spin excitations. Our studies reveal the low-temperature magnetic phase diagram and enable us to measure characteristic critical properties. We compare our results closely with current theoretical models, and this gives some further insight into the nature of the spin-liquid phase.