A rather heavy hand has reworked the tune above.

The song is in the first issue of Stevenson and Moore's 'A Selection of Irish Melodies' and the tune there is given as below in 'Sources of Traditional Irish Music', 1998. (Original edition of 1st 8 of 10 parts are in the Library of Congress, but I failed to copy the tune from part 1).

A variant of the tune is that for "The Boys of Kilkenny". (In DT without tune.) M. Kelly printed single sheet songs from 1802-11, and it isn't known for certain which of these two versions of the tune was published earliest. The single sheet issue of "The Boys of Kilkenny" is in the Folger Shakespeare Library. Note after the first verse the tune changes at the "X" to a different ending.

X:1
T:The Meeting of the Waters, tune- THE OLD HEAD OF DENNIS
S:'A Selection of Irish Melodies', part 1, #12 (1807)
Q:120
L:1/8
M:6/8
K:A
e3/4d/4|c3/2B/A AFE|EFAA2B3/4c/4|ddc/B/ BcA|ecA B2 B/c/|\
ddc/B/ BcA|ecA{c/}B2 A3/4B/4|B3/2B/A A/F3/2E|EFA d2d/c/|\
{c/}BBA A/F3/2E|EFAA2|]

^^ X:2
T:THE BOYS OF KILKENNY (1st verse)
S:Single sheet song issued by Michael Kelly, c 1805
Q:120
L:1/8
M:6/8
K:Bb
f3/4 e/4|d3/2 c/ B A F F|G B B B2 B/ e/|e3/2 d/ c e d B|\
f3/2d/ d/ B/c2e|e3/2 d/ c e3/2 d/ B|f3/2 d/ B/ d/ .c2 B/ B/|\
d3/2 c/ B B G F|G B B .e2 "X"f3/4 e/4|d3/2 c/ B B G F|\
G B B B2|]

X:3
T:The Boys of Kilkenny (all verses but 1st)
Q:120
L:1/8
M:6/8
K:Bb
f3/4 e/4|d3/2 c/ B A F F|G B B B2 B/ e/|e3/2 d/ c e d B|\
f3/2d/ d/ B/c2e|e3/2 d/ c e3/2 d/ B|f3/2 d/ B/ d/ .c2 B/ B/|\
d3/2 c/ B B G F|G B B .e2 "X"e3/4 d/4|\
d3/4 c/4 c3/4 B/4 B3/4 G/4G2 F|G B B B2|]