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"Today's outcome is a historic step on the road to economic recovery," Kenny told the Irish parliament.
Kenny said the agreement would ensure repayments were reduced and spread over a longer period, reducing pressure on Ireland, which is struggling to recover despite an EU-IMF bailout in 2010.
Ireland was determined to lessen the burden from a 31-billion-euro ($41.9-billion) promissory note -- effectively a high-interest IOU that was pumped into Anglo Irish to rescue it during the financial crisis.
Kenny told parliament that with interest, the cost of the deal over its lifetime would have been almost 48 billion euros -- they were "a highly onerous and unfair legacy of the banking crisis".
He said: "I am pleased to announce that today Ireland has reached a conclusion to its discussions with the European Central Bank that delivers on our commitment to put in place a fairer and more sustainable arrangement.
"Under the agreement... the promissory notes are being exchanged for long-term Irish government bonds with maturities of up to 40 years."
In the early hours of Thursday, Irish lawmakers voted through emergency legislation to liquidate Anglo Irish, which was nationalised in 2009 and is now known as the Irish Bank Resolution Company.
Kenny said the deal for Anglo Irish and the former Irish Nationwide Building Society "bookends a tragic chapter in our country's history" and had removed "stains on our international reputations and dents to our national pride".
The bank's assets will be transferred to Ireland's state-run "bad bank", the National Asset Management Agency (NAMA).
Earlier in Frankfurt, ECB chief Mario Draghi said the board had "unanimously noted" the Irish proposals.
The issue of promissory notes has dominated the Irish political landscape in recent months, with many opposition figures calling for the government to default on the next 3.1 billion euros payment that was due in March.
Opposition lawmakers slammed the proposals in a heated debate.
"You tell us you are going to replace the promissory note with a sovereign bond (with) no write-down whatsoever," Sinn Fein's finance spokesperson Pearse Doherty said in response to Kenny.
"You have said to every single man, woman and child in this state that they will pay back every single penny of the toxic Anglo debt.
"This is not the deal the Irish people were waiting for," he added.
Ireland's once-proud 'Celtic Tiger' economy, famed for its double-digit growth for a decade from the mid-1990s, came close to collapse as it was hit by a property market meltdown, soaring state debt and high unemployment.
In 2010, Anglo Irish posted a 17.65 billion euro loss, the largest corporate loss in Irish history.
Dublin was forced to accept an 85 billion euro bailout from the European Union and the International Monetary Fund in November 2010 after reckless lending by banks brought the economy to its knees.
Ireland hopes to become the first bailed-out eurozone nation to exit its rescue programme by returning fully to the sovereign markets by the end of 2013.
The government and the IMF had argued that a deal on the promissory notes would increase the attractiveness of Irish bonds and push down the yields making it more likely for a smooth transition from the bailout programme.