It's a 78 rpm record I listened to as a child in the early 50's. It
was a story about a horse who had a bell in his heart. The horse's
name was Fantisimo. I think the singer/narrator was Betty Hutton, but
I'm not sure.
How could I find a copy of this record or at least the story?

You were close in guessing Betty Hutton; it was actually Betty Garrett
who narrated the story.

"Fantissimo, the Musical Horse" was one of three stories on the
record. The other two stories (about a bear and a raindrop) were
narrated by Keenan Wynn.

"...it reminds me of an album I had from the 50's or early 60's. Three
stories: 'Drippy, the Runaway Raindrop,'... 'The Bear That Wasn't
(name of the album),' and 'Fantissimo, the Musical Horse.' First two
narrated by Keenan Wynn. The title story, with great sound effects,
was about a bear who's accidently stuck in a factory, and mistaken for
'a silly man who needs a shave and wears a fur coat.' The bosses and
even other bears refuse to believe he's a bear."

Loganberry Books
http://www.logan.com/loganberry/stump-cd.html

"L70076 - The Bear That Wasn't/Drippy/Fantissimo [1958]" 

Lion Album Discography 
http://www.bsnpubs.com/mgm/lion50.html

"CH-113 - The Bear That Wasn't/Drippy (The Runaway
Raindrop)/Fantissimo - Keenan Wynn [1962] Reissue of Lion L70076."

MGM Album Discography
http://www.bsnpubs.com/mgm/mgmchildren.html

This was a tough one to track down, but I have found one copy for
sale. The cost is $6, plus shipping. This appears to be the LP
reissue, rather than the 78rpm original:

"MGM
P310 NC0109 CH 113 Tashin: The Bear that Wasn't; Shelton: Drippy (The
Runaway Raindrop).Keenan WYNN (narr). J. Jackson: Fantissimo (The
Musical Horse). Betty GARRETT (narr). A, LSM $6"

Parnassus Records
http://www.parnassusrecords.com/p310.htm

Contact and ordering information for Parnassus Records may be found on
the page linked above.

Google search strategy:

Google Web Search: fantissimo horse
://www.google.com/search?hl=en&q=fantissimo+horse

I hope this helps! If anything is unclear or incomplete, please
request clarification; I'll be glad to offer further assistance before
you rate my answer.

Best regards,
pinkfreud

Thank you guys! I never expected the answer so fast. I've already
ordered the item online. Fantastic! Or maybe I should say
Fantissimo!!!
Sincerely, Kathy E