Go to the on-campus clinic the day of the exam. Make sure to mention to the doctor that you have an exam that day and he'll fill out a form. When you get back from the clinic send an email to the course director explaining the situation. You don't need to include details (e.g. symptoms).
A business day or two later you'll get an email from the Dean of Students office saying that they received your medical excuse. Then someone will be in touch with when you need to take the completion exam. For a final exam it is at the start of the next term.
It appears the school policy has changed somewhat and the course director can chose to give a cumulative or essay completion exam, presumably to scare of people who fake illness to get additional study time.