So I took down my handy-dandy micro-print Oxford English Dictionary (2d ed.) and was surprised to find that the word "rat" in all its variations takes up about two whole pages.

There on page 1,512 (yeah, really) I found the noun form of the slang, there defined as: "A police informer; an informer in a prison." It proceeds to give examples of early usage, the first being dated 1902.

More on point, the verb form appears on the next page and is defined as: "To desert one's party, side, or cause, esp. in politics; to go over as a deserter; to turn traitor. Also, in Criminals' slang, to inform." An instance dated 1812 includes the construction "ratted upon."

And finally, there's this: "With on. To default on; to let (someone) down; to behave disloyaly towards; [snip] to inform on." For this, they quote instances that appear as early as 1932.

All that, and still nothing on the specific phrase "rat out." Sorry.