  For $p = 3,5,7,13$, we consider a ${\mathbb Z}_p$-orbifold construction of
the Moonshine vertex operator algebra $V^\natural$. We show that the vertex
operator algebra obtained by the ${\mathbb Z}_p$-orbifold construction on the
Leech lattice vertex operator algebra $V_\Lambda$ and a lift of a
fixed-point-free isometry of order $p$ is isomorphic to the Moonshine vertex
operator algebra $V^\natural$. We also describe the relationship between those
${\mathbb Z}_p$-orbifold constructions and the ${\mathbb Z}_2$-orbifold
construction in a uniform manner. In Appendix, we give a characterization of
the Moonshine vertex operator algebra $V^\natural$ by two mutually orthogonal
Ising vectors.
