Hi, Stephen - you're right that it's by Dorothy Fields and Jerome Kern. It's listed by the Harry Fox Agency (search for the exact title A Fine Romance), and you can get a recording license at http://www.songfile.com/ - the process is fairly easy for small quantities - all you need is your trusty old credit card.

Here is the copyright gibberish, from Reading Lyrics, by Robert Gottlieb and Robert Kimball (published in 2000):

A Fine Romance ©1936 (Renewed), Universal-Polygram International Publishing, Inc. and Aldi Music (c/o Songwriters Guild of America). All rights for U.S. and Canada controlled by Universal-Polygram International Publishing, Inc. All rights reserved. Warner Brothers Publications U.S., Inc., Miami, FL 33014

Hope that does the trick for you. It's a fine song, isn't it? There's a playful recording by Ella Fitzgerald and Louis Armstrong that brings out all the humor of the lyrics.

-Joe Offer-

Lyrics at Rabid Squirrel's Jazz Archive.

Ella: https://www.youtube.com/watch?v=u92sIAN3rxk

Ella & Louis: https://www.youtube.com/watch?v=KTBUgzdE_Ro