The problem of solidification by a line heat sink in an infinite medium with cylindrical symmetry for a substance having an extended freezing temperature range between the solidus and liquidus temperatures is solved exactly for two different cases characterizing the distribution of the solid fraction within the two-phase zone. In one of the models, the solid fraction is assumed to vary linearly with the temperature and in the other solidification within the two-phase zone is assumed to have a linear relationship with the distance. The analysis is applicable for both eutectic and solid solution alloys.