Answer:
Both forms princess' and princess's are correct.
Step-by-step explanation:
In the English language, when a singular noun ends in -s, we should add 's to it in order to form its possessive. There have always been some exceptions, such as ancient names and the possessive Jesus'. Thus, even though the word princess ends in -s, it's correct possessive would be princess's.
However, more and more authors agree that pronunciation should be taken into consideration. If the extra -s after the apostrophe sounds awkward, the best would be to leave out and only add '. Some grammar authors don't even worry about pronunciation, they only claim that both spellings are acceptable (e.g. Diana Hacker in "A Writer's Reference"). In conclusion, princess' is also a correct way to spell the word's possessive form.