We theoretically study a Gaussian optical beam with an embedded off-axis optical vortex. We also experimentally generate such an asymmetric Gaussian optical vortex by using an off-axis spiral phase plate. It is shown that depending on the shift distance the laser beam has the form of a crescent, which is rotated upon propagation. An analytical expression is obtained for the orbital angular momentum of such a beam, which appears to be fractional. When the shift increases, the greater the number of spirality of the phase plate or the "fork" hologram, the slower the momentum decreases. The experimental results are in qualitative agreement with the theory.