It is widely believed that andesitic magmas erupted at arc-volcanoes are stored in shallow reservoirs prior to eruption, but high-resolution images of focused regions of magma in the shallow crust are rare. We integrate seismic tomography with numerical models of magma chamber growth to constrain the magma chamber beneath Soufrière Hills Volcano, Montserrat. Our approach reveals the characteristics and dynamics of the magmatic system with a level of detail that no single method has yet achieved. The integrated analysis suggests that a magma chamber of 13 km3 with over 30% melt fraction formed between 5.5 and at least 7.5 km depth, a significantly higher melt fraction than inferred from the seismic data alone. The magma chamber may have formed by incremental sill intrusion over a few thousand years and is likely to be a transient, geologically short-lived feature. These volume and geometry estimates are critical parameters to model eruption dynamics, which in turn are key to hazard assessment and eruption forecasting.