The three-dimensional Ising model on the m$\times$n$\times$l cubic lattice with the screw boundary condition along the ${\bf X}$ direction and the periodic boundary conditions along both ${\bf Y}$ and ${\bf Z}$ directions is exactly solved by using the $2^{mn}$-dimensional representation of the rotation group in $2mn$-dimensions, similar to the Kaufman's spinor approach in two dimensions. The exact partition function is obtained from two sets of $2^{mn-1}$ eigenvalues of the $2^{mn}$-dimensional transfer matrix {\bf V} corresponding to the even and odd eigenvectors, respectively. Such the eigenvalues are determined by the angles of the 2mn-dimensional rotation associated with {\bf V}.