The proposed algorithm is based on the block anti-triangular form of the original matrix M, as introduced by the authors in [11]. Via successive orthogonal similarity transformations this form is then updated to a new form , whereby the first k rows and columns of have elements bounded by a given threshold τ and the remaining bottom right part of is maintained in block anti-triangular form. The updating transformations are all orthogonal, guaranteeing the backward stability of the algorithm, and the algorithm is very economical when the near rank deficiency is detected in some of the anti diagonal elements of the block anti-triangular form. Numerical results are also given showing the reliability of the proposed algorithm.