A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter h of the cross-section tends to zero. More precisely, we show that stationary points of the nonlinear elastic functional E^h, whose energies (per unit cross-section) are of order h^2, converge to stationary points of the Gamma-limit of E^h/h^2. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James and Mueller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.