For the first time, an analytical solution has been derived for Stokes flow through a conical diffuser under the condition of partial slip. Recurrent relations are obtained that allow determination of the velocity, pressure and stream function for a certain slip length λ. The solution is analysed in the first order of decomposition with respect to a small dimensionless parameter ${\lambda }/{r}$. It is shown that the sliding of the liquid over the surface of the cone leads to a vorticity of the flow. At zero slip length, we obtain the well-known solution to the problem of a diffuser with a no-slip boundary condition corresponding to strictly radial streamlines. To solve that problem, we use an alternative form of the general solution of the linearised, stationary, axisymmetric Navier–Stokes equations for an incompressible fluid in spherical coordinates. A previously published solution to this problem, dating back to the paper by Sampson (1891 Phil. Trans. R. Soc. A, vol. 182, pp. 449–518), is given in terms of a stream function that leads to formulae that are difficult to apply in practice. By contrast, the new general solution is derived in the vector potential representation and is simpler to apply.