The magnetohydrodynamic stability of a free‐boundary straight plasma column with a slightly elongated rectangular cross section, carrying a uniform axial current, is investigated analytically. Axisymmetric modes (n= 0), for which the circular cross section case is marginally stable (m= 1), are considered. Stability is found for a rigid vertical displacement, in good agreement with numerical results. However, a vertical displacement with a small additionalm= 3 like perturbation, localized near the plasma surface, leads to instability.