In an earlier research paper [HL1], we presented a novel, yet straightforward linear-time algorithm for merging two sorted lists in a fixed amount of additional space. Constant of proportionality estimates and empirical testing reveal that this procedure is reasonably competitive with merge routines free to squander unbounded additional memory, making it particularly attractive whenever space is a critical resource. In this paper, we devise a relatively simple strategy by which this efficient merge can be made stable, and extend our results in a nontrivial way to the problem of stable sorting by merging. We also derive upper bounds on our algorithms' constants of proportionality, suggesting that in some environments (most notably external file processing) their modest run-time premiums may be more than offset by the dramatic space savings achieved.