This paper deals with a two dimensional Stefan problem of solidification in arbitrary bounded domains. To realize this, the governing equations are transformed from a complex physical domain to a simpler computational domain using body-fitted coordinates and the corresponding transformed equations are solved using an alternating direction implicit (ADI) method. Stability analysis of the proposed ADI scheme, applied to the transformed equations, is discussed using von Neumann method. The developed scheme is then used to solve problems which are initially defined in some regular geometries like rectangular or circular but changes with time to some complex shapes. Comparisons with the existing schemes show that the developed scheme produced better solutions.