Water molecules in the narrow cylindrical pore of a (6,6) carbon nanotube form single-file chains with their dipoles collectively oriented either up or down along the tube axis. We study the interaction of such water chains with homogeneous electric fields for finite closed and infinite periodically replicated tubes. By evaluating the grand-canonical partition function term-by-term, we show that homogeneous electric fields favor the filling of previously empty nanotubes with water from the bulk phase. A two-state description of the collective water dipole orientation in the nanotube provides an excellent approximation for the dependence of the water-chain polarization and the filling equilibrium on the electric field. The energy and entropy contributions to the free energy of filling the nanotube were determined from the temperature dependence of the occupancy probabilities. We find that the energy of transfer depends sensitively on the water-tube interaction potential, and that the entropy of one-dimensionally ordered water chains is comparable to that of bulk water. We also discuss implications for proton transfer reactions in biology.