We consider the problem of computing the solution of large-scale discrete ill-posed problems when there is noise in the data. These problems arise in important areas such as seismic inversion, medical imaging and signal processing. We pose the problem as a quadratically constrained least squares problem and develop a method for the solution of such problem. Our method does not require factorization of the coefficient matrix, it has very low storage requirements and handles the high degree of singularities arising in discrete ill-posed problems. We present numerical results on test problems and an application of the method to a practical problem with real data.