Law School

Re impossibility: not true. Most universities only prefer students from their LLM going into their SJD programs, although you do have some which make it a compulsory requirement. The main thing however is the strength of YOUR application, proposal and Letters of recommendation. I know someone who went into Georgetown's SJD program without being in their LLM - she had a strong topic as well as a LOR from one of the law school's professors.

Edit: And of course, there are the other requirements such as good academic history, employment history, personal statements - what comprises of the LLM application.