The Taoiseach Micheál Martin has described Barry Cowen's statement to the Dáil for drink-driving as a 'clear abject apology'.
A spokesperson for Mr Martin said it is clear the Agriculture Minister deeply regrets what happened and accepts that his remorse is genuine.
Mr Cowen told the Dáil it was a "stupid, stupid mistake".
Minister Cowen also said it was "not uncommon" for people of all ages to drive with either a provisional licence or learners' permit at the time.
He told deputies on Tuesday he wanted to "officially put on the record" the facts of the incident.
He confirmed he had consumed two drinks before attending the All Ireland Football Final in Dublin in 2016, before eating a light meal after the match.
While driving home to Offaly, he was stopped and breathalysed.
After being found to be over the legal alcohol limit, he was given a €200 fine and a three-month driving ban.
He held a learner drivers' permit at the time as his previous provisional licence had lapsed.
He said: "Before the recent and proper reforms of the system, it was not uncommon for people of all ages and level of experiences to drive with either a provisional or learners' permit. However, this was bad practice - and I clearly should have regularised my position much sooner."
After reports of previous parking and speeding fines, Minister Cowen said he has examined his records and found the only additional, unreported event he was able to identify was "a failure to display a parking disc while parked in Tullamore 14 years ago".
In terms of the driving ban, he said: "My decision in September 2016 to drive home after consuming any alcohol was a stupid, stupid mistake... it had never happened before the 18th September 2016, and it has never happened since.
"It is a mistake for which I am profoundly sorry.
"This grave error and my subsequent humiliation will hopefully to highlight the terrible dangers and consequences of drink driving."
But Newstalk's political correspondent Sean Defoe has said he is not sure a line has been drawn under the controversy.
"He didn't actually address a lot of the issues that people want answered last night.
"Indeed, some of the road advocate groups said that it raised more questions than answers, particularity in relation to the idea of why he was on a learners permit provisional licence in the fist place.
"He didn't address that - just saying he hadn't regularised his position before it".