JINGLE BELLS in the DT is said to have been composed by John Pierpont in 1859. The original, however, was written and composed by "J[ames]. Pierpont", titled "The One Horse Open Sleigh", and published in 1857. There're some differences in lyrics & melody. Here's the sheet music from the Levy collection (also HERE at American Memory).

Title: The One Horse Open Sleigh. Song and Chorus.
Composer, Lyricist, Arranger: Written and Composed by J. Pierpont[typo corrected].
Publication: Boston: Oliver Ditson & Co., 277 Washington St., 1857.
Form of Composition: strophic with chorus
Instrumentation: piano and voice
First Line: Dashing thro' the snow in a one horse open sleigh
First Line of Chorus: Jingle bells, Jingles bells, jingle all the way; Oh! what joy it is to ride
Dedicatee: To John P. Ordway, Esq.
Plate Number: 18200
Subject: Sleds & sleighs
Subject: Transportation
Subject: Courtship & love
Subject: Accidents
Call No.: Box: 062 Item: 044

THE ONE HORSE OPEN SLEIGH
(J. Pierpont) [James Pierpont]
(MIDI from Public Domain Music: 19th Century American Popular Music)

1.
Dashing thro' the snow,
In a one horse open sleigh,
O'er the hills we go,
Laughing all the way;
Bells on bob tail ring,
Making spirits bright,
Oh what sport to ride and sing
A sleighing song to night.

CHORUS:
Jingle bells, Jingle bells,
Jingle all the way;
Oh! what joy it is to ride
In a one horse open sleigh.
Jingle bells, Jingle bells,
Jingle all the way;
Oh! what joy it is to ride
In a one horse open sleigh.

2.
A day or two ago
I tho't I'd take a ride
And soon miss Fannie Bright
Was seated by my side,
The horse was lean and lank
Misfortune seem'd his lot
He got into a drifted bank
And we, we got upsot.

(CHORUS)

3.
A day or two ago,
The story I must tell
I went out on the snow
And on my back I fell;
A gent was riding by
In a one horse open sleigh,
He laughed as there I sprawling lie,
But quickly drove away.

(CHORUS)

4.
Now the ground is white
Go it while you're young,
Take the girls to night
And sing this sleighing song;
Just get a bob tailed bay
Two forty as his speed,
Hitch him to an open sleigh
And crack, you'll take the lead.

(CHORUS)

Robert DeCormier Singers & Ensemble (on A Victorian Christmas) sing "The One Horse Open Sleigh" based on the original (HERE is their sound clip). The song was published later as "Jingle Bells, or, The One Horse Open Sleigh" in 1859.

Title: Jingle Bells, or, The One Horse Open Sleigh. Song & Chorus.
Composer, Lyricist, Arranger: By J. Pierpont.
Publication: Boston: Oliver Ditson & Co., 277 Washington St., 1857*.
Form of Composition: strophic with chorus
Instrumentation: piano and voice
First Line: Dashing thro' the snow, In a one horse open sleigh
First Line of Chorus: Jingle bells, Jingle bells, jingle all the way. Oh! what joy it is to ride
Plate Number: 23488
Subject: Sleds & sleighs
Subject: Snow
Subject: Transportation
Subject: Courtship & love
Call No.: Box: 062 Item: 029

*The date should have been "1859", according to Richard Jackson (Popular Music of Nineteenth-Century America, Dover, p. 272) and James J. Fuld (The Book of World-Famous Music, p. 313).

"The song is said to have been originally written for a local Sunday-school entertainment. [...] James Pierpont was born in Boston in 1822, became a composer and died in Winter Haven, Fla., in 1893. James Pierpont was a son of John Pierpont, the grandfather of John Pierpont Morgan, the banker and founder of The Pierpont Morgan Library. [...] The earliest known printing of the familiar chorus was in Students' Songs, published and arranged by Harvard Students, and copyrighted June 9, 1880,..., and there is a comma after 'Jingle'" (James J. Fuld, The Book of World-Famous Music, 4th ed., Dover, 1995, pp. 313 & 680). The titles (now with the familiar chorus) in The Most Popular College Songs (Hinds, Hayden & eldridge, 1904, 1906, pp. 74-75) and Carmina Princetonia: The Princeton Song Book (G. Schirmer, 1914, 1927, p. 121) are "Jingle, Bells!" and "Jingle, Bells" respectively (with no mention of the composer).

~Masato