It's almost identical to the tune of Banks of the Bann.

BANKS OF THE BANN


When first to this country a stranger I came
I placed my affection in a girl that was young
She being fair and tender, her waist small and slender
'Twas nature that formed her for my overthrow.

On the banks of the Bann, it was there I first met her
She appeared like an angel or Egypt's fair queen
Her eyes were like diamonds or stars brightly shining
She's one of the fairest in the world that I've seen

Oh it was her cruel parents that first caused a variance
Because she is rich and above my degree
But I do endeavor to gain my love's favor
Although she is come of a high family

oh, me name it is Delaney, it's a name that don't shame me
And if I'd saved money I'd never have roamed
But drinking and sporting, night rambling and courting
Are the cause of all my ruin and me absence from home

But had I the wealth that is in the Indies
I'd put rings on her fingers and gold in her ears
And there on the banks of the lovely Bann River
In all kinds of splendor I'd live with my love.

A friend of mine says the "Slane" tune is also related to a song called "With my love in the Road."
^^

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See also The Banks of the Bann, a substantially different version of this song.
-Joe Offer-