Nanopore desalination technology hinges on high water-permeable membranes which, at the same time, block ions efficiently. In this study, we consider a recently synthesized [Science363, 151-155 (2019)] phenine nanotube (PNT) for water desalination applications. Using both equilibrium and non-equilibrium molecular dynamics simulations, we show that the PNT membrane completely rejects salts, but permeates water at a rate which is an order-of-magnitude higher than that of all the membranes used for water filtration. We provide the microscopic mechanisms of salt rejection and fast water-transport by calculating the free-energy landscapes and electrostatic potential profiles. A collective diffusion model accurately predicts the water permeability obtained from the simulations over a wide range of pressure gradients. We propose a method to calculate the osmotic water permeability from the equilibrium simulation data and find that it is very high for the PNT membrane. These remarkable properties of PNT can be applied in various nanofluidic applications, such as ion-selective channels, ionic transistors, sensing, molecular sieving, and blue energy harvesting.