Freezing of liquids in one dimension is studied by a lattice model that is an extension of the model solvent of the hydrophobic attraction. The model in one dimension, which is exactly solvable, exhibits a continuous phase change between a high-temperature disordered "liquid" state and a low-temperature ordered "solid" state but also does exhibit a first-order freezing transition at some finite temperature with either one of the two model parameters taken to be infinite. In this theoretical framework the sharpness of the freezing in one dimension is expressed by a simple function of the microscopic model parameters and thus is related with other macroscopic properties of the substance. These results may account for continuity and discontinuity of the liquid and solid reported for different one-dimensional substances. (C) 2003 American Institute of Physics.