The two-sided Hamburger moment problem, also called the strong one, has been extensively studied in recent years in connecton with rational approximation. Here the author considers the question of when a sequence, say \(\{a_ n\}^ \infty_{n=0}\) can be extended backwards so that the resulting sequence \(\{a_ n\}^ \infty_{n=-N}\) has an integral representation of Hamburger type. This was settled earlier (without proof) by him [C. R. Acad. Sci., Paris, Sér. I 292, 431-432 (1981; Zbl 0457.47034)] under different circumstances. In this paper he discusses the problem completely, as well as the possibility of extending \(\{a_ n\}^ \infty_{n=0}\) to \(\{a_ n\}^ \infty_{n=-\infty}\).