The author presents a pseudospectral (or collocation) approximation of the unsteady Stokes equations. Using the Uzawa algorithm, the spectral system is decoupled into Helmholtz equations for the velocity components and an equation with the pseudo-Laplacian for the pressure. Both the velocity components and the pressure are approximated by polynomials of the same degree \(N\). The author constructs compatible velocity and pressure approximations to ensure that the discrete problem is well-posed. The resulting discrete velocity field is identically divergence-free. Only one grid (no staggered grids) with standard Chebyshev Gauss-Lobatto nodes is used. Further the author compares his treatment with a non-homogeneous Neumann condition for the pressure proposed in [G. E. Karniadakis, M. Israeli and S. A. Orszag, J. Comput. Phys. 97, No. 2, 414-443 (1991; Zbl 0738.76050)]. The comparison shows that the Neumann condition yields somewhat better approximations for the pressure. In the time discretization, a higher-order backward differentiation scheme for the intermediate velocity is combined with a higher-order extrapolant for the pressure. It is shown numerically that a stable third-order method in time can be achieved. In the case of periodic conditions, this result is confirmed by Fourier analysis.