"He must have come across the information about the Nights some time between 25 February and 13 October 1701. Muhsin Mahdi writes about the acquisition of the G-Manuscript: It was published in Europe under the title Tausendundeine Nacht. In 2004 Claudia Ott translated this manuscript (using Mahdi’s edition) into German. In 1990 WW Norton published an English translation by Husain Haddawy of Mahdi’s recension titled The Arabian Nights consisting only of these oldest stories. This set is expensive when found but many major university libraries have a copy or have access to one. His out of print three volume set from Brill consists of the manuscript, several essays in English, several indexes in Arabic including an intensive comparative index and a descriptive chapter of old manuscripts of the Nights and an introduction in Arabic of his theories regarding the various origins of the Nights. In 1984 Muhsin Mahdi, a professor at Harvard, published an edited and collated version of this Arabic manuscript (along with a completed version of “Qamar al-Zaman”) in which he attempts to portray the oldest and most authentic Nights as anyone knew them. The last story is known generally (though has several different spelling and other title variants) as “The Tale of Qamar al-Zaman and Budur” and does not have an ending. Galland had requested that his colleagues look for the complete or original Nights after he translated a stand-alone version of “Sindbad” (aka “Sinbad the Sailor and Sinbad the Porter” or several other related sounding/spelled titles) from Arabic into French that someone told him was a part of a larger body of work (the Nights). A friend seems to have brought from Syria, but was in France with them when Galland acquired them. Galland received the Arabic manuscript volumes in 1701 while in France. (Thanks to JC for the dates of publication). These volumes were in Antoine Galland’s personal library and he appeared to use them as the primary (but far from only) source for the beginning volumes of his French translation of the Nights which were titled Mille et une Nuits (1704-1717). I've never read about a certain title on this manuscript nor are the stories individually titled (can someone verify or refute this?) yet they are segmented into numbered “nights.” On the picture above however each section clearly says "Alf Layla wa Layla" in Arabic. There are 282 “nights” and around 35 stories. It’s not clear who the author or scribe is (yet! maybe you can find out) or exactly where or when or how this manuscript came into being (besides allegedly being sent from Syria to Galland in 1701). This date is disputed however and some think it is written earlier in the 14th century. It is the oldest manuscript of the Nights in any language that contains any stories and people have dated it to somewhere around the 15th century AD. Its call numbers at the library are MSS arabes 3609, 36.
The often-called “Galland manuscript” is an Arabic language manuscript of the Nights in the Bibliotheque nationale in Paris France.