Abstract A one-dimensional model for ice accretion due to incoming supercooled water impacting on a conducting substrate is developed, where the substrate is cooled from below by a liquid or gas. Both rime and glaze ice situations are considered. Non-dimensionalisation shows that conduction is the dominant method of heat transfer and so the heat equations are reduced to pseudo-steady forms. In this case the problem reduces to solving a single equation for the ice layer thickness. The water height and temperatures in the ice, water and substrate may subsequently be found. The asymptotic solution is validated by comparison with results from a numerical scheme which solves the full Stefan problem. This is an extension of a previously published solution method that involved simpler boundary conditions. For glaze ice, a comparison including water droplet energy either in the boundary conditions or as a source term in the heat equations, is also performed.