Newly elected Ceann Comhairle Verona Murphy has said she is "humbled, honoured and privileged" to serve in the role.
Ms Murphy, an independent TD from Wexford, was confirmed as elected to the role after a majority of TDs voted in her favour.
She becomes the first woman to hold the role, and her election paves the way for a new government made up of Fianna Fáil, Fine Gael and the Regional Independent Group.
Ms Murphy was elected on the third count when transfers from outgoing ceann comhairle Seán Ó Fearghaíl were distributed, pushing her over the quota of 87.
On the first count she was ahead with 76 votes, but got no transfers when the 21 votes for Fianna Fáil's John McGuinness were distributed on the second count.
At that stage Aengus Ó Snodaigh of Sinn Féin was on 54 votes and Mr Ó Fearghaíl was on 42.
Ms Murphy picked up 13 votes from Mr Ó Fearghaíl on the third count, which put her over the line.
Ms Murphy committed to be fair and impartial as well as "even-handed and even-tempered" in the role as Ceann Comhairle, and said she would improve her Irish "to a level that is commensurate with the office I now hold".
"To the people of Wexford, I want to say my office is open and normal service resumes after the Christmas holidays, and will be maintained and improved."
She was applauded as she crossed the chamber floor to assume the role, with Taoiseach Simon Harris and Tanaiste Micheal Martin reaching over to shake her hand.
Ms Murphy then walked across to shake hands with Sinn Féin leader Mary Lou McDonald and Labour leader Ivana Bacik, as well as other TDs before thanking her team, supporters and the regional group of independents.
Speaking in the chamber, Fine Gael leader Simon Harris congratulated Ms Murphy on becoming Ceann Comhairle.
"I know you will serve with diligence, with impartiality and with decency," he said.
"That is a long-awaited milestone moment in Irish politics and I sincerely wish you all of the best as you discharge your duties," he said.
Despite resigning this morning, Mr Harris continues to serve as Taoiseach until a successor is appointed at the conclusion of government formation talks. He will attend a European Council meeting in Brussels tonight.
The Cabinet will also remain in office.
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Verona Murphy: By-election failure to Ceann Comhairle
The first sitting of the 34th Dáil got under way this morning with Dáil clerk Peter Finnegan reading out the names of the 174 TDs.
Among them were 64 new TDs including the youngest member of the house, 24-year-old Labour TD for Cork North-Central Eoghan Kenny. The longest-serving member of the Dáil is Fianna Fáil's Wille O'Dea, who was first elected in 1982.
The candidates in the race to become the next Ceann Comhairle were given five minutes each to make their case in the Dáil.
Mr McGuinness spoke of how TDs rather than party leaders should decide the matter.
Ms Murphy told TDs of her intention to uphold simple principles to ensure deputies could represent their electorate
Mr Ó Fearghaíl vowed to continue his work to reform the Dáil, while Mr Ó Snodaigh stressed his Irish language qualifications and even-handedness.
McDonald nominated as Taoiseach, Martin critical of move
Sinn Féin leader Mary Lou McDonald was nominated as Taoiseach in the Dáil by two of her party colleagues, however the motion was defeated by 110 votes to 44.
The party's finance spokesperson Pearse Doherty said that during the election the party had argued Ireland needed a new government and a new leader, and said "Mary Lou McDonald is that leader".
"The government Teachta McDonald would lead would be a government of change, a new government that would end the housing crisis and make housing affordable for working people, a government that would transform our health services and end the wait for treatment, one that would introduce affordable childcare for families, ensure a better deal for ordinary workers and ensure our young people can build a future and a life here in Ireland," he said.
He added: "We've had 100 years of Fine Gael and Fianna Fail governments. It is enough."
Calling Ms McDonald a leader "of incredible strength, resilience and perseverance", he said there was a need "to break with the merry-go-round of a century".
In her maiden speech to the Dáil, Sinn Féin TD for Louth Joanna Byrne seconded Ms McDonald's nomination.
The Sinn Féin motion was defeated by 44 votes for, 110 against, with 14 abstentions.
Fianna Fáil leader Micheál Martin criticised Sinn Féin for nominating Ms McDonald as the next taoiseach.
He said that after the 2020 general election, Sinn Féin argued the party that received the largest vote should lead the government.
"So, if it was being consistent, it would be in the process of nominating me today," he said, prompting jeers in the Dáil.
He interjected amid noise in the chamber that "breaking out the flags and celebrating an exit poll is more than a bit premature".
"I think one of the things which most encourages cynicism about politics is when our time is wasted on empty gestures and grandstanding," Mr Martin said.
"There is no better way to describe the decision by Sinn Féin to propose its leader for the position of taoiseach today, and we all know that no-one has the support yet to be elected as taoiseach today. There's no new government available yet."
