If there's no current copyright, no one needs to authorizes usage.

My guess, however, is that someone does hold a copyright on some recorded arrangement or adaptation. Possibly the best known was recorded by Ian & Sylvia about 1962.

If they sued, St. Martins would presumably be able to show that at least one version of the song with the words you want is truly in the PD.

This is something you should take up with them. It may be that the quotation is so trivial that no copyright holder could object with any expectation of gain (particularly of you don't include music).