Molecular dynamics simulations were performed to investigate the transport and phase change of a water cluster confined in a single-walled carbon nanotube. It is demonstrated that the water cluster can be transported along the temperature gradient due to the temperature dependence of the interactive potential energy of water molecules. The efficiency of the transport takes advantage of the hydrophobic water-carbon interface. Furthermore, the first order phase change of the water cluster to ice-nanotube is investigated in terms of the dependence of the freezing temperature on the diameter of the carbon nanotube. It is shown that the freezing temperature exhibits a maximum value, about room temperature, for nanotube diameter of d=1.1nm, where the ice-nanotube consists of 5-membered rings. Analyses of the potential energy indicate that the freezing temperature is mainly determined by the structural stability of the ice crystal.