I'm planning on taking Bio 1A03 and Chem 1A03 in the spring session this year. The session ends on Friday the 19th of June. Does anyone know when exams for these courses are usually held in the spring session?

They are held on the very last day of your class, in your classroom or one of the testing rooms on campus and they are supervised by the instructor.

Final exams days in summer school are usually assigned by the instructor him/herself. As the above post said, it is usually done on the very last day of class.