This paper deals with the solution of the pure Neumann problem for the diffusion equation by a finite element method. First, an extended generalized formulation of the Neumann problem in the Sobolev space H1(Ω) is derived and investigated. Then a discrete analog of this problem is formulated by using standard finite element approximations of the space H1(Ω). An iterative method for solving the corresponding SLAE is proposed. Some examples of solving model problems are used to discuss the numerical properties of the algorithm proposed.