In this paper we study perturbations of the stiffly weighted pseudoinverse (W1/2 A)+W1/2 and the related stiffly weighted least squares problem, where both the matrices A and W are given with W positive diagonal and severely stiff. We show that the perturbations to the stiffly weighted pseudoinverse and the related stiffly weighted least squares problem are stable, if and only if the perturbed matrices (^)A = A+δA satisfy several row rank preserving conditions.