This paper reports a new physical mechanism that enables the heating of ions by a low-frequency parallel propagating Alfven wave of finite amplitude in a low beta plasma. The heating does not rely on ion cyclotron resonance. The process has two stages: First, ions, whose initial average velocity is zero, are picked up in the transverse direction by the Alfven wave and obtain an average transverse velocity. Second, at a given location the parallel thermal motions of ions produce phase differences (randomization) between ions leading to the heating of ions. The randomization (or heating) process saturates when phase differences are sufficiently large. The time scale over which ions are significantly heated is ∼π (k vth) (vth is the initial ion thermal speed and k is the wave number). The heating is dominant in the perpendicular (to the background magnetic field) direction. Subsequently, a large ion temperature anisotropy is produced. During the heating process, ions are also accelerated in the parallel direction and obtain a bulk flow speed along the background magnetic field.