Magnetization reversal in a ferromagnetic nanowire which is much narrower than the exchange length is believed to be accomplished through the thermally activated growth of a spatially localized nucleus, which initially occupies a small fraction of the total volume. To date, the most detailed theoretical treatments of reversal as a field-induced but noise-activated process have focused on the case of a very long ferromagnetic nanowire, i.e., a highly elongated cylindrical particle, and have yielded a reversal rate per unit length, due to an underlying assumption that the nucleus may form anywhere along the wire. But in a bounded-length (though long) cylindrical particle with flat ends, it is energetically favored for nucleation to begin at either end. We indicate how to compute analytically the energy of the critical nucleus associated with either end, i.e., the activation barrier to magnetization reversal, which governs the reversal rate in the low-temperature (Kramers) limit. Our treatment employs elliptic functions, and is partly analytic rather than numerical. We also comment on the Kramers prefactor, which for this reversal pathway does not scale linearly as the particle length increases, and tends to a constant in the low-temperature limit.