Follow along here, at Breviary.net. It's a beautiful chant, isn't it? It's 10 full minutes - the YouTube limit, I believe - and you don't get a bit tired of it. The cantor has a beautiful voice; I remember thinking so as I watched it live. (Even though I have no particular fondness for John Paul II - he said and did some very harmful things while he was Pope, when it comes to the lives of gay people - I do like the still image on the YouTube video link. Very dramatic and quite beautiful; in fact, I suspect I would have liked him a lot - except for the virulent homophobia he seemed always compelled to express.)
Note, too: the Litany from the 1662 Book of Common Prayer looks to be very similar to the later sections of the Litany of the Saints. The English Reformers dropped the prayers to the saints, but retained the Supplications. I'm actually not sure how far back this chant goes, or how long it's been in its present form; I'll do a little research on that and post again.
EDIT: Here's New Advent on the Litany of the Saints, which it says is "of great antiquity":
It was used in the "Litania Septiformis" of St. Gregory the Great, and in the procession of St. Mamertus. In the Eastern Church, litanies with the invocation of saints were employed in the days of St. Basil (d. 379) and of St. Gregory Thaumaturgus (d. about 270) (Basil, Ep. lxiii; Socrates, VI, viii, Sozomen, VIII, vii). It is not known when or by whom the litany was composed, but the order in which the Apostles are given, corresponding with that of the Canon of the Mass, proves its antiquity (Walafr. Strabo, "De Reb. Eccl.", xxiii).
Much more at the link.