We predict the maximum solidification rate, or critical velocity, V-c at which an insoluble particle suspended in a melt is pushed ahead of an advancing solidification front by intermolecular forces. At higher solidification rates the particle is incorporated within the solid. The net intermolecular force pushing the particle and the viscous resistance opposing it are both significantly influenced by the shape of the front as it conforms to the particle in response to interfacial premelting. We predict the entire shape of the front, within a thin-film approximation, accounting for the freezing-point depression due to curvature. We show how the interface shape varies with the magnitude of the surface energy and the closeness of the particle, and compare these to previous, ad hoc representations of the interface. We confirm the scaling results of previous, more approximate, analyses for the case in which the intermolecular forces are dominated by nonretarded van der Weals interactions, and provide new results for other power-law interactions. We examine how the particle behaviour changes as its radius increases so that the effect of interfacial curvature is diminished.