The theories of partial metric spaces and of weightable quasi-metric spaces are equivalent, as was shown by \textit{S. G. Matthews} [Ann. New York Acad.Sci. 728, 183--197 (1994; Zbl 0911.54025)]. The present authors prove that the bicompletion of a weightable quasi-metric space is weightable. Thus, every partial metric space has a partial metric completion that is unique up to isometry. As examples, it is shown that the completion of the partial metric space of finite words is the space of at most countable words, and that the partial metric space of complexity functions is a completion of eventually constant complexity functions.