We develop a physical model of the thermal history of the ureilite parent body (UPB) that numerically tracks the history of its heating, hydration, dehydration, partial melting and smelting as a function of its formation time and the initial values of its composition, formation temperature and water ice content. Petrologic and chemical data from the main group (non-polymict) ureilite meteorites, which sample the interior of the UPB between depths corresponding to pressures in the range 3–10 MPa, are used to constrain the model. We find that to achieve the 30% melting inferred for ureilites from all sampled depths, the UPB must have had a radius between 80 and 130 km and must have accreted about 0.55 Ma after CAI formation. Melting began in the body at 1 Ma after CAI, and the time at which 30% melting was reached varied with depth in the asteroid but was always between 4.5 and 5.8 Ma after CAI. The total rate at which melt was produced in the UPB varied from more than 100 m3 s−1 in the very early stages of melting at 1 Ma after CAI to 5 m3 s−1 between 2 and 3 Ma after CAI, decreasing to extremely small values as the end of melting was approached beyond 5 Ma. Although the initial period of high melt production occupied only a short time around 1 Ma after CAI, it corresponded to half (16%) of total silicate melting, and all strictly basaltic (i.e. plagioclase-saturated) melts must have been produced during this period. A very efficient melt transport network, consisting of a hierarchy of veins and larger pathways (dikes), developed quickly at the start of melting, ensuring rapid (timescales of months) transport of any single parcel of melt to shallow levels, thus ensuring that chemical interaction between melts and the rocks through which they subsequently passed was negligible. Volatile (mainly carbon monoxide) production due to smelting began at the start of silicate melting in the shallowest parts of the UPB and at later times at greater depths. Except at the very start and very end of melting, the volatile content of the melts produced was always high – generally between 15 and 35 mass % – and most of the melt produced was erupted at the surface of the UPB with speeds well in excess of the escape velocity and was lost into space. However, we show that 30% melting at the 3 MPa pressure level was only possible if 15% of the total melt produced in the asteroid was retained as a small number (5) of very extensive, sill-like intrusions centered at a depth of 7 km below the surface, near the base of the 8 km thick outer crust of the asteroid that was maintained at temperatures below the basalt solidus by conductive heat loss to the surface. The horizontal extents of these sills occupied about 75% of the surface area of the UPB, and the sills acted as buffers between the steady supply of melt from depth and the intermittent explosive eruption of the melt into space. We infer that samples from these intrusions are preserved as the rare feldspathic (loosely basaltic) clasts in polymict ureilites, and show that the cooling histories of the sills are consistent with these clasts reaching isotopic closure at 5 Ma after CAI, as given by 26Al–26Mg, 53Mn–53Cr and Pb–Pb age dates.