Mr. Hubert Beckwith was so kind to contact me; he provided me with new information, an addition to the webpage Stabat Mater text. On this page Hans has written about the origins of Stabat Mater text:  …The oldest reference in a liturgical source is a Breviary from Arezzo, a manuscript dated 1417. ..
That seems to be incorrect now! Mr. Hubert wrote to me:

…Not long after your husband died, an earlier liturgical use of Stabat Mater was discovered. Here is a short article in English that describes: the discovery of the famous Stabat Mater being used as a sequence in the Gradual produced by a convent of Dominican nuns in Bologna in the later thirteenth century. This is by far the earliest known manuscript example of this hymn used as a sequence rather than as a devotional hymn. It has been commonly believed that the hymn only became used as a sequence in the late middle ages. (Its melody there is not found in other sources.)  The article links to a research study by Cesarino Ruini (in Italian) that discusses the discovery in great detail.  Follow the link “scarica questo file PDF.”  I hope you can find someone to translate it.
As a brief summary: The sentences about early sources might be changed to something like “The Stabat mater first appeared as a liturgical sequence in the later 13th century, in a Gradual from a Dominican convent in Bologna. Since the beginning of the 14th century it can be found in prayer books.
Again, many thanks to you and your late husband for your richly informative and elegant website…

I wil make a correction to the webpage as soon as possible. Thank you very much Mr. Beckwith!