We have performed atomistic molecular dynamics (MD) simulations of water molecules inside narrow, open-ended carbon nanotubes placed in a bath of water molecules. The radius of the tube is such that only a single file of water molecules is allowed inside the tube. The confined water molecules are shown to be positionally ordered even at a temperature of 300 K. The calculated mean-square displacement (MSD) of the confined water molecules reveals that initially the water molecules undergo ballistic motion that crosses over to normal (Fickian) diffusion at longer times. We also develop a random-walk model in 1 D for the motion of a cluster of water molecules inside the nanotube. The agreement of the MSD calculated from the MD simulation and from the 1D random-walk model establishes the occurrence of normal diffusion of water molecules even in a tube where single-file diffusion is expected.