This blog took part in some discussion about possibilities for an immersion course in an ancient language (basically Latin or Greek) in 2014 with the following results.
The obvious answer would be something in Latin, and there did indeed turn out to be such things in the US: Conversational Latin Seminars at the University of Kentucky in Lexington and, with a Roman Catholic connection, Latin Summer Immersion at Wyoming Catholic College, Cenaculum Sancti Hieronymi in Mobile, Alabama. (Now you can see some despatches from Lexington here.)
Outside of the US, Accademia Vivarium Novum is an eight-week intensive Latin course in Rome (with the possibility of some Greek as well). Staying in Rome, the Pontifical University of the Holy Cross offers intensive summer courses in Latin, Greek (Koine) and Biblical Hebrew. Now they seem to be connected with the Polis Institute in Jerusalem, who have summer courses in Latin and Syriac. They also have a textbook, Polis : Parler le grec ancien comme une langue vivante, as well as Latin and (Koine) Greek courses in Barcelona and Latin and Greek courses in Florida.
Having apparently abandoned Israel, the Biblical Language Centre will be doing a Biblical Hebrew/Koine Greek Ulpan in North Carolina.
Finally, the Paideia Institute offers Living (Medieval) Latin in Paris, Living Latin in Rome and Living Greek in Greece (and they also do telepaideia, online courses in conversational Latin and Attic). Living Greek in Greece is an intensive introduction to spoken to Attic Greek. In two seminar-style meetings per day, participants read and discuss ancient Greek literature and philosophy in Attic Greek. Each year, readings are organized around a set theme. The theme for 2014 is Dionysus. The readings are Euripides’ Bacchae and the Homeric Hymns to Dionysus. Now that sounds well exciting, like something out of Donna Tartt…At the same venue, one can also Speak and philosophize in Ancient Greek (scroll some way down the page!)