We generalize the CGME (Conjugate Gradient Minimal Error) algo-rithm to the weighted and regularized least squares problem. Analysis ofthe convergence of generalized CGME and CGLS shows that CGMEcanbe expected to perform better for ill-conditioned regularization matrices.Two different types of regularization are considered: anℓ1penalty andanℓ2penalty. Theℓ1problem is solved using Iterative Reweighted LeastSquares, which leads to an ill-conditioned regularizationmatrix. The twomethods are applied in a low-field MRI framework. The MRI physics ina low-field scanner are simulated to generate a noisy signal.When anℓ1penalty is used and iterative reweighted least squares isemployed, GCGLS needs significantly more iterations to converge thanGCGME. GCGME has a regularizing effect that leads to fewer artifactsin our simulations. This effect seems to be stronger when a lower numberof CG iterations is used. These two observations indicate that GCGMEis a very promising alternative to GCGLS.