I'm surprised that the play was apparently never revived in the 60s-80s by some veteran movie actress performing on the dinner theater or summer stock circuit. It would have been the perfect vehicle for a glamorous older actress. What fun it would have been to see Bette Davis, Gloria Swanson, Myrna Loy, Dorothy Lamour, Lana Turner, or one of the other movie legends who was doing theatre in this period in the part.

Many thanks for these links. I have wanted to read the original stories since I first saw the 1930 and 1959 versions of the movie.

Yes, the 1959 film
The Bat
was based on the stage play of the same name, which was written by Mary Roberts Rinehart and Avery Hopwood. This play was first produced on Broadway in August of 1920:
http://www.ibdb.com/show.php?id=1862
The play was based on Miss Rinehart's novel
The Bat
, written with Avery Hopwood, which was published in 1920.
Miss Rinehart also published a novel called
The Circular Staircase
in 1908.
Both of these novels, like most of Miss Reinhart's books, are in the public domain and may be read online or downloaded free.
The Bat
:
http://www.publicbookshelf.com/romantic-suspense/the-bat/
http://www.gutenberg.org/etext/2019
The Circular Staircase
:
http://www.publicbookshelf.com/romance/circular-stair/
http://www.gutenberg.org/etext/434
General information:
http://en.wikipedia.org/wiki/Mary_Roberts_Rinehart
http://www.publicbookshelf.com/author/Mary-Roberts-Rinehart
http://www.gutenberg.org/browse/authors/r#a183

Okay, got it. This should clear things up:
"In 1926, a novelization of 'The Bat' appeared, apparently written by poet Stephen Vincent Bent with little input from Rinehart. This novel version usually appears in paperback under Rineharts name, without any mention of Hopwood [Rinehart's co-dramatist/collaborator on the original play] or Bent."
Source: http://www.mysteryfile.com/blog/?p=125
Incidentally, the year 1926 was also the year of the flamboyant Roland West film version, so that was probably the impetus for commissioning the above-referenced novelization. And I'm sure it's quite good, too, since Bent (author of, e.g., "The Devil and Daniel Webster") had a knack with the sinister.

It was written in 1926 and it is a separate story than that of the Circular Staircase, similiar they are in certain elements but there are obvious differences.

Okay, if you've read "The Circular Staircase" and Rinehart's play "The Bat," you can see the relationship between the two, but also how greatly expanded and improved the play is.
If, however, you have a novel called "The Bat," could you provide the information from the flyleaf, copyright and credit pages? What year was it written? Is it merely a re-issue of "The Circular Staircase" under the later, more popular title? (Though that would be a little deceptive, since the character of The Bat doesn't actually appear in the earlier novella.) Or is it, as someone else has suggested, merely a novelization of the play and/or the movie by some hired hand in the 50's or 60's? I still contend there is no utterly separate prose (i.e., non-dramatic) work called "The Bat" which was actually penned by Mary Roberts Rinehart herself.
Please advise!

I also remember reading the book. In the fifties and sixties it was common to publish novelizations of popular films as a promotional tool. These books were, sometimes loosely, based on the screenplay.
Hope that clears things up.

You say there's no such book but I find that a little hard to believe as I own both the Ciruclar Staircase, AND the Bat, both novels or novellas if you prefer, and by this time I have also read the play The Bat so I know full well that the two do exist separately, or in this case the three of them do.

You said you've read the novel "The Bat," but there really was no such novel. The original Mary Roberts Rinehart novella (1908) was called "The Circular Staircase" (not to be confused with "the Spiral Staircase," a good Dorothy McGuire thriller from the 40s). Anyway, Rinehart herself later adapted the novella, with the help of writing partner-dramatist Avery Hopwood, and greatly expanded the original plot, even adding the character of the Bat, and called the play just that "The Bat" (1920). The play, as another commenter has noted, was indeed highly successful through the 20s and is, in fact, far better either than the original novella OR this dopey Moorehead/Price vehicle. Matter of fact, it was so successful that after Rinehart made a bundle selling rights for a silent film version, a lawsuit was commenced over the EARLIER screen rights to the underlying and inferior novella.
The original 1926 silent film, incidentally, is quite flamboyant, but really interesting, and much more imaginative than this 1959 reductio ad absurdum.
Anyway, the Rinehart stage version has lots of atmosphere, snappy dialogue, special surprise effects, and is still popular among amateur and school groups. Performance rights are still leased through Samuel French, Inc.

The play was popular in the twenties. I don't think its been performed in decades.