For possible solving of axisymmetric Stokes-flow boundary value problems in spheroidal coordinates of arbitrary given focal distance, the authors have derived a general semiseparable solution for the Stokes stream function expressed as an infinite series in terms of products of Gegenbauer polynomials of the first and second kind. This is done first by finding the separable solution of \(E^ 2\Psi=0\) (\(E^ 2\) is the Stokes operator for axisymmetric flow in spheroidal coordinates) and then by finding the complete class of solutions \(\tilde\Psi\) for which \(E^ 2\tilde\Psi\) is an eigenfunction of \(E^ 2\Psi=0\). All these infinitely many functions \(\Psi\) and \(\tilde\Psi\) alone and their linear composition satisfy the Stokes equation \(E^ 4\Psi=0\). As the focal distance tends to zero, the general solution reduces to the known separable solution of the problem in spherical coordinates. As an example, the solution of the spheroid-in-cell problem is treated. Note the missing minus sign in equation \((4)\).