"Sumer Is Icumen In," words and music still together, dates from about 1260 according to the best authorities. Gaudeamus Igitur in original form supposedly dates to the same time. The words now sung to the latter are relatively new (18th century). Surely the English song is the oldest one still sung regularly, same words, same tune.   
An image of the original "Sumer..." is on the web (link Sumer Manuscript, above, Dec 01, 02, 2:50 pm). Is an image of the Gaudeamus..." Ms. available?