Since Orszag's paper ['Accurate solution of the Orr -Sommerfeld stability equation ', J. Fluid Mech., 50, 689 -703 (1971)], most of the subsequent spectral techniques for solving the Orr -Sommerfeld equation (OSE) employed the Tau discretization and Chebyshev polynomials. The use of the Tau discretization appears to be accompanied by so-called spurious eigenvalues not related to the OSE and a singular matrix B in the generalized eigenvalue problem. Starting from a variational formulation of the OSE, a spectral discretization is performed using a Galerkin method. By adopting integrated Legendre polynomials as basis functions, the boundary conditions can be satisfied exactly for any spectral order and the non-singular matrices A and B are obtained in Ax =uBx. For plane Poiseuille flow, the stiffness and the mass matrices are sparse with bandwidths 7 and 5 respectively, and the entries can be calculated explicitly (thus avoiding quadrature errors) for any polynomial flow profile U. According to the convergence results [Hancke, 'Calculating large spectra in hydrodynamic stability: a p FEM approach to solve the Orr-Sommerfeld equation', Diploma Thesis, Swiss Federal Institute of Technology Zu ¨rich, Seminar for Applied Mathematics, 1998; Hancke, Melenk and Schwab, 'A spectral Galerkin method for hydrodynamic stability problems', Research Report No. 98-06, Seminar for Applied Mathematics, Swiss Federal Institute of Technology, Zu ¨rich], no spurious eigenvalue has been found. Numerical experiments with spectral orders up to p=600 illustrate the results.