Sparse linear least squares problems containing a few relatively dense rows occur frequently in practice. Straightforward solution of these problems could cause catastrophic fill and delivers extremely poor performance. This paper studies a scheme for solving such problems efficiently by handling dense rows and sparse rows separately. How a sparse matrix is partitioned into dense rows and sparse rows determines the efficiency of the overall solution process. A new algorithm is proposed to find a partition of a sparse matrix which leads to satisfactory or even optimal performance. Extensive numerical experiments are performed to demonstrate the effectiveness of the proposed scheme. A MATLAB implementation is included.