A pivoting strategy for the Gaussian elimination of linear systems whose coefficients matrices are sign-regular is proposed. An \(n\times m\) matrix \(A\) is sign-regular if, for each \(k\) \((1\leq k\leq\min \{n,m\})\), all \(k\times k\) submatrices of \(A\) have a determinant with the same nonstrict sign. Backward error analysis of this pivoting strategy is performed and small error bounds are obtained.