Many extensions of the standard model of particle physics suggest that neutrinos should be Majorana-type fermions--that is, that neutrinos are their own anti-particles--but this assumption is difficult to confirm. Observation of neutrinoless double-β decay (0νββ), a spontaneous transition that may occur in several candidate nuclei, would verify the Majorana nature of the neutrino and constrain the absolute scale of the neutrino mass spectrum. Recent searches carried out with 76Ge (the GERDA experiment) and 136Xe (the KamLAND-Zen and EXO (Enriched Xenon Observatory)-200 experiments) have established the lifetime of this decay to be longer than 1025 years, corresponding to a limit on the neutrino mass of 0.2-0.4 electronvolts. Here we report new results from EXO-200 based on a large 136Xe exposure that represents an almost fourfold increase from our earlier published data sets. We have improved the detector resolution and revised the data analysis. The half-life sensitivity we obtain is 1.9 × 1025 years, an improvement by a factor of 2.7 on previous EXO-200 results. We find no statistically significant evidence for 0νββ decay and set a half-life limit of 1.1 × 1025 years at the 90 per cent confidence level. The high sensitivity holds promise for further running of the EXO-200 detector and future 0νββ decay searches with an improved Xe-based experiment, nEXO.