The celebrated Poincaré (1910) steady solution, the uniform vorticity flow in a precessing spheroid, which is realized in the inviscid interior region outside the boundary layer in the strong spin and weak precession limit, is derived, as a unique solution analytically without any prior assumption on the spatial structure, and its singular behavior for a sphere is resolved. Assuming that the spin and precession axes are respectively parallel and perpendicular to the symmetry axis of the spheroid, we denote the spin and precession angular velocities by Po / 2.620 δ 2 + ( 0.258 5 δ + 1 - c ) 2 , respectively. The latter is included in Busse (1968)'s formula, whereas the former is new. An analysis of the same flow as the present paper from a different approach was performed recently by Zhang et al (2014). Unfortunately, however, an inconsistency is found in their results.