It has been suggested that the presence of frazil ice can lead to a conditional instability in seawater. Any frazil forming in the water column reduces the bulk density of a parcel of frazil-seawater mixture, causing it to rise. Due to the pressure-decrease in the freezing point, this causes more frazil to form, causing the parcel to accelerate, and so on. We use linear stability analysis and a non-hydrostatic ocean model to study this instability. We find that frazil ice growth caused by the rising of supercooled water is indeed able to generate a buoyancy-driven instability. Even in a gravitationally stable water column, the frazil ice mechanism can still generate convection. The instability does not operate in the presence of strong density stratification, high thermal driving (warm water), a small initial perturbation, high background mixing or the prevalence of large frazil ice crystals. In an unstable water column the instability is not necessarily expressed in frazil ice at all times; an initial frazil perturbation may melt and refreeze. Given a large enough initial perturbation this instability can allow significant ice growth. A model shows frazil ice growth in an Ice Shelf Water plume several kilometres from an ice shelf, under similar conditions to observations of frazil ice growth under sea ice. The presence of this instability could be a factor affecting the growth of sea ice near ice shelves, with implications for Antarctic bottom water formation.