The doubly constrained robust Capon beamformer (DCRCB), which employs a spherical uncertainty set of the steering vector together with the constant norm constraint, can provide robustness against arbitrary array imperfections. However, its performance can be greatly degraded when the uncertainty bound of the spherical set is not properly selected. In this paper, combining the DCRCB and the weight-vector-norm-constrained beamformer (WVNCB), we suggest a new robust adaptive beamforming method which allows us to overcome the performance degradation due to improper selection of the uncertainty bound. In WVNCB, its weight vector norm is limited not to be larger than a threshold. Both WVNCB and DCRCB belong to a class of diagonal loading methods. The diagonal loading range of WVNCB, which dose not consider negative loading, is extended to match that of DCRCB which can have a negative loading level as well as a positive one. In contrast to the conventional DCRCB with a fixed uncertainty bound, the bound in the proposed method varies such that the weight vector norm constraint is satisfied. Simulation results show that the proposed beamformer outperforms both DCRCB and WVNCB, being far less sensitive to the uncertainty bound than DCRCB.