Estimating the values of unknown parameters in ill-posed problems from corrupted measured data presents formidable challenges in ill-posed problems. In such problems, many of the fundamental estimation methods fail to provide meaningful stabilized solutions. In this work, we propose a new regularization approach combined with a new regularization-parameter selection method for linear least-squares discrete ill-posed problems called constrained perturbation regularization approach (COPRA). The proposed COPRA is based on perturbing the singular-value structure of the linear model matrix to enhance the stability of the problem solution. Unlike many regularization methods that seek to minimize the estimated data error, the proposed approach is developed to minimize the mean-squared error of the estimator, which is the objective in many estimation scenarios. The performance of the proposed approach is demonstrated by applying it to a large set of real-world discrete ill-posed problems. Simulation results show that the proposed approach outperforms a set of benchmark regularization methods in most cases. In addition, the approach enjoys the shortest runtime and offers the highest level of robustness of all the tested benchmark regularization methods.