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A finite element method is proposed to solve a scalar singular diffusion problem. The method is constructed by adding to the standard Galerkin method a mesh-dependent term obtained from a least-squares form of the gradient of the Euler-Lagrange equation. For the one-dimensional homogeneous problem the method is designed to develop a nodal exact solution. An error estimate shows that the method converges optimally for any value of the singular parameter. Numerical results demonstrate the good stability and accuracy properties of the method.