We model the solidification and subsequent cooling of a supercooled liquid droplet that is lying on a cold solid substrate after impact. It is assumed that solidification occurs for a given fixed droplet shape. The shapes used by the model are a sphere, truncated spheres, and an experimentally registered droplet shape. The freezing process is conduction-dominant and is modeled as a one-phase Stefan problem. This moving boundary problem is reformulated with the enthalpy method and then solved numerically with an implicit finite-difference technique. The numerical results for the simple case of a spherical droplet touching a surface are similar to those of a freely freezing spherical droplet and are well confirmed by the ID asymptotic analytical model of Feuillebois et al. (J. Colloid Interface Sci. 169 (1995) 90). A freezing water droplet is considered as an example. The numerical results for full freezing time, subsequent cooling time, and last freezing point coordinate for the various droplets shapes are fitted by analytical functions depending on supercooling, thermal resistance of the target surface (expressed by Biot number), and spreading parameter. These functions are proposed for direct application, thus avoiding the need to solve the full freezing and cooling problem. (C) 2003 Elsevier Inc. All rights reserved.