A one-dimensional model for ice growth in the presence of an incoming supercooled fluid is developed by specifying a Stefan problem. A number of assumptions are made, relevant to relatively thin ice and water layers, which allow the thermal problem to be solved exactly in terms of the layer thicknesses and ambient conditions. The ice thickness is then determined by combining the mass balance with the phase change condition to reduce the problem to a single first-order ordinary differential equation, which requires a numerical solution. Quantities such as the water film thickness and temperature profiles in the ice and water can be determined easily once the ice thickness is known. Results are presented showing ice and water growth, subject to various ambient conditions. Qualitative agreement with experiment is also shown and the implications of the results to the physical process discussed.