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The University of Chicago Law School is now offering a fully funded (including living stipend) one year M.L.S. degree, allowing a student to spend a year here taking courses, seminars and workshop in the law school that will complement and facilitate your dissertation work. The application information is here and more information about the program is here: Download MLS Degree Info Sheet Updated 7 24 15. At the time of application you should be done with or finishing your coursework and embarking on a dissertation proposal for which deeper knowledge of law would be helpful. Feel free to shoot me an e-mail if you have questions not answered by the preceding documents. Note that if you decide you want to earn a J.D., you will not be able to count any of your MLS coursework towards the JD (this is an American Bar Association rule, not a rule of our law school).
There is no other law school in the United States presently offering such a degree.