The phase conversion of amorphous solid water (ASW) to crystalline ice (CI) has been investigated in the very thin (similar to 10 monolayers) film regime on a Ru(0 0 1) surface. We analyze the converted CI fraction with the Avrami model, and recognize that one-dimensional CI growth occurs, which can be contrasted to the three-dimensional CI growth generally established in the thick (>= 50 monolayers) film regime. We evaluate activation energy for the ASW crystallization to be about 1.0 eV. We suggest that the ASW crystallization is not influenced by the substrate even near the substrate-ice interface. Crown Copyright (C) 2009 Published by Elsevier B.V. All rights reserved.