As Snuffy so rightly deduced the song referred to is indeed Master McGrath, however, I fear that in popularising folk music the Clancy's did seem to lose a verse here and there.

Here follows a very slightly different and more complete version along with chords that Noel Murphy has sung for many many years
Chords don't appear to have uploaded real accurately but should give you some idea

MASTER McGRATH

       Am                        G           Am
  Eighteen sixty-nine being the date of the year,
         C         Em           Am
  Those Waterloo sportsmen and more did appear
                     Em                   Am
  To win the great prizes and bear them awa'
         Em         Am           Em        Am
  Never counting on Ireland and Master McGrath.

On the twelfth of December, that year of renown,
McGrath and his keepers they left Lurgan town.
A gale in the channel it soon drove them o'er.
On the thirteenth, they landed on England's fair shore.

And when they arrived there in big London town,
Those great English sportsmen all gathered around,
And some of them sneered with a scornful haw-haw,
Saying: “Is that the great dog you call Master McGrath?”

And one of these gents from his nose looking down,
Said: “We don't give a curse for your Irish greyhound.”
And another one sneered with a scornful haw-haw,
Saying: “We'll humble the pride of your Master McGrath.”

Lord Lurgan stepped forward and said: “Gentlemen,
If there's any amongst you with money to spend,
For your great English nobles we don't give a straw.
Here's five thousand to one upon Master McGrath.”

Well, McGrath he looked up and he wagged his old tail
Informing his lordship: “I know what you mean.
Don't fear them, dear Lurgan; don't heed them at all.
I'll soon cripple their laughter,” says Master McGrath.

There stood Rose of England, the Saxon's great pride.
The Master quite easy stood close by her side.
They led them away and the crowd cried “Hurrah
For the pride of all England and Master McGrath!”

As Rose and the Master they both ran along,
“I wonder,” said Rose, “what brought you from your home?
You should have stayed home in your Irish domain
And not come hunting laurels on Albion's plain.”

“Now I know,” says McGrath, “we have wild heather bogs,
But old Ireland is famous for good men and dogs.
Lead on, bold Britannia; give none of your jaw.
Stuff that up your nostrils,” says Master McGrath.

The hare she ran on with a wonderful view
And swift as a rocket they crossed Waterloo.
Rose gave the first turning according to law,
But the second was given by Master McGrath.

McGrath paced the hare just as fast as the wind.
He was sometimes before it and sometimes behind;
Then he jumped on its back and he held up his paw.
“Long live auld Ireland!” says Master McGrath.