A solution of the Navier-Stokes equations is obtained for the flow resulting from the steady rotation of a semi-infinite right circular (solid) cylinder about its vertical axis. Incompressible viscous fluid is assumed to fill the space outside the cylinder on one side of a horizontal solid plane. In the proposed method of solution the pertinent physical quantities are expressed as series in positive powers of the Reynolds number Re with space-dependent coefficients. It is shown that the coefficients of (Re)M can be obtained by solving linear partial differential equations which depend on the coefficients of (Re)i, where i < M. A truncated solution, which holds for small Re, is obtained by solving for the first two coefficients. These results indicate that at the flat end of the cylinder the pressure distribution is nearly constant, yet along the adjacent bounding plane it rises with the radial direction.