Ok, since I wanted to know the answer too, I tried it. So, it turns out that as long as you're in a distinguished program when you graduate, you get the distinguished degree. My sim earned 11 credits in regular Fine Arts at Foxbury, and for her final term, transferred to Britechester and took one class in Fine Arts. She got her distinguished degree, and got a job as a musician at level 8 to start.
Just remember that you have to stay in the same major, or you'll lose progress. If she had chosen a different major, she would have needed 7 credits to graduate instead of just 1.