The traditional singular value decomposition (SVD) method is unable to diagnose the weak fault feature of bearings effectively, which means, it is difficult to retain the effective singular components (SCs). Therefore, a new singular value decomposition method, SVD based on the FIC (fault information content), is proposed, which takes the amplitude characteristics of fault feature frequency as the selection index FIC of singular components. Firstly, the Hankel matrix of the original signal is constructed and SVD is applied in the matrix. Secondly, the proposed index FIC is used to evaluate the information of the decomposed SCs. Finally, the SCs with fault information are selected and added to obtain the denoised signal. The results of bearing fault simulation signals and experimental signals show that compared with the traditional differential singular value decomposition (DS-SVD), the proposed method can select the singular components with larger amount of fault information, and is able to diagnose the fault under the heavy noise interference. The new method can be used for signal denoising and weak fault feature extraction.