The Meissner solution of a smooth cylindrical superconducting domain subject to a uniform applied axial magnetic field is examined. Under an additional convexity condition the uniqueness of the Meissner solution is proved. It is then shown that it is a local minimiser of the Ginzburg-Landau energy E_kappa. For applied fields less than a critical value the existence of the Meissner solution is proved for large enough Ginzburg-Landau parameter kappa. Moreover it is proved that the Meissner solution converges to a local minimiser of a certain energy E_infinity in the limit as kappa tends to infinity. Finally, it is proved that for kappa large enough the Meissner solution is not a global minimiser of E_kappa.