Some of Earth's largest magmatic provinces result from the interaction between mid-ocean ridges and near-ridge hotspots, which are hypothesized to overlie plumes of upwelling mantle. Geodynamic models predict that upwelling plumes are sheared by the motion of the overlying tectonic plates and can connect to a nearby mid-ocean ridge by shallow flow beneath thin, young lithosphere. Here we present seismic tomographic images of the upper 300 km of the mantle beneath the Galápagos Archipelago in the eastern Pacific Ocean. We observe a low-velocity anomaly, indicative of an upwelling plume, that is not deflected in the direction of plate motion. Instead, the anomaly tilts towards the mid-ocean ridge at depths well below the lithosphere. These characteristics of the plume-ridge connection beneath the Galápagos Archipelago are consistent with the presence of multiple stages of partial melting, melt extraction, and melt remixing within the plume and surrounding mantle. These processes affect the viscosity of the asthenosphere, alter the upwelling plume and influence the compositions of surface lavas. Our results imply that the coupling between the oceanic plate and plume upwelling beneath the Galápagos is weak. Multistage melting may similarly affect the geophysical and geochemical characteristics of other hotspots.