Recently,Wei in[18]proved that perturbed stiff weighted pseudoinverses and stiff weighted least squares problems are stable,if and only if the original and perturbed coefficient matrices A and A satisfy several row rank preservation conditions.According to these conditions,in this paper we show that in general,ordinary modified Gram-Schmidt with column pivoting is not numerically stable for solving the stiff weighted least squares problem.We then propose a row block modified Gram-Schmidt algorithm with column pivoting,and show that with appropriately chosen tolerance,this algorithm can correctly determine the numerical ranks of these row partitioned sub-matrices,and the computed QR factor R contains small roundoff error which is row stable.Several numerical experiments are also provided to compare the results of the ordinary Modified Gram-Schmidt algorithm with column pivoting and the row block Modified Gram-Schmidt algorithm with column pivoting.