We examine the effect of a strong shear flow on morphological instabilities that occur in the directional solidification of a dilute, binary alloy when the interface departs from local thermal equilibrium in a frozen-temperature, one-sided model. In particular, the flow velocity U∞ is much larger than the rate of solidification V and the Schmidt number is arbitrary. In contrast to solidification processes under small or no flow, for which both a cellular and an oscillatory mode of instability appear, the liquid-solid interface under flows of large magnitude is susceptible to a single mode of instability. All experiments on banding occur in the overlap region between these modes under no flow and since these two primary modes coalesce into one time-dependent, spatially-dependent mode, there is no mechanism for the production of bands anymore in the large-flow regime. No experiments have been conducted to date on this rapid solidification under flow and this suggests that such experiments would show no bands. The flow significantly stabilizes and selects higher wavenumbers for the preferred form of instability in comparison to systems involving small or no flow. The introduction of a strong flow provides an effective mechanism to eliminate instabilities at high solidification rates. However, stability thresholds at low solidification rates remain mainly indifferent to the presence of flow.