The under-abundance of very massive galaxies(1,2) in the Universe is frequently attributed to the effect of galactic winds(3-6). Although ionized galactic winds are readily observable, most of the expelled mass (that is, the total mass flowing out from the nuclear region) is likely to be in atomic(7,8) and molecular phases(9-11) that are cooler than the ionized phases. Expanding molecular shells observed in starburst systems such as NGC 253 (ref. 12) and M 82 (refs 13, 14) may facilitate the entrainment of molecular gas in the wind. Although shell properties are well constrained(12), determining the amount of outflowing gas emerging from such shells and the connection between this gas and the ionized wind requires spatial resolution better than 100 parsecs coupled with sensitivity to a wide range of spatial scales, a combination hitherto not available. Here we report observations of NGC 253, a nearby(15) starburst galaxy (distance similar to 3.4 megaparsecs) known to possess a wind(16-20), that trace the cool molecular wind at 50-parsec resolution. At this resolution, the extraplanar molecular gas closely tracks the Ha filaments, and it appears to be connected to expanding molecular shells located in the starburst region. These observations allow us to determine that the molecular outflow rate is greater than 3 solar masses per year and probably about 9 solar masses per year. This implies a ratio of mass-outflow rate to star-formation rate of at least 1, and probably similar to 3, indicating that the starburst-driven wind limits the star-formation activity and the final stellar content.