Summary: Truncated singular value decomposition is a popular method for solving linear discrete ill-posed problems with a small to moderately sized matrix \(A\). Regularization is achieved by replacing the matrix \(A\) by its best rank-\(k\) approximant, which we denote by \(A_k\). The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L-curve criterion. This paper describes a novel regularization approach, in which \(A\) is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as \(A_k\). Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of \(A\) by \(A_k\).