The Ulster Bank has applied to join the Irish government's scheme to guarantee all the deposits in Irish banks.
The legislation necessary for the move received final approval from the Dáil (Irish parliament) on Thursday.
The Ulster Bank - a subsidiary of the Royal Bank of Scotland - said it was confident of qualifying.
Irish ministers had said they may extend the scheme to foreign banks with a significant number of Irish customers on a case-by-case basis.
The Seanad (upper house) passed the bill on Thursday morning, but because of some amendments, it had to go back to the Dáil (lower house) for final approval before receiving the presidential signature.
In the course of the Dáil debate, Finance Minister Brian Lenihan confirmed representatives of the public interest will be appointed to the boards of institutions accepting the support.
He said the Irish state was "getting deep" into the banking system, and must ensure that the taxpayer would be protected.
Competition
Mr Lenihan said he would also take action to prevent what he called "excessive risk taking" being rewarded in the remuneration of top executives.
The Dáil sat until after 0200 BST, the latest sitting in three decades, to debate the legislation, finally backing it by 124 votes to 18, with only Labour opposed.
The government's 400bn euro guarantee covers Allied Irish Bank, Bank of Ireland, Anglo Irish Bank, Irish Life & Permanent, Irish Nationwide Building Society and the Educational Building Society.
With the scheme also covering the institutions' branches in the UK - some opponents of the move have said it will unfairly make the Irish banks more attractive than their British counterparts.
The European Commission has said it is "in close contact" with the Irish government about the move following concern that it may give Irish banks an unfair advantage over foreign competitors.
The British Bankers' Association said that the Republic of Ireland's move had put British banks at a competitive disadvantage.
There are signs that some UK savers are considering moving their money to Ireland's banks.
The Bank of Ireland has said it has seen a steady increase in people contacting its call-centres.
The Post Office, whose savings products are backed by the Bank of Ireland, has also reported a rise in the number of customers seeking to invest.
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