Connolly was given the presidential seal of office during her inauguration ceremony in Dublin Castle on Tuesday afternoon.
Taoiseach Micheal Martin congratulated the Galway native with an address to St Patrick’s Hall.
Martin said: "On this solemn and joyful day of celebration, it is a great honour for me to congratulate President Catherine Connolly on behalf of the Government and the people of Ireland on entering upon her office as the President of Ireland."
The Taoiseach also thanked former President Michael D. Higgins for his "outstanding contribution" during his time in office.
He said the country owed him and his wife, Sabina, a "tremendous debt of gratitude".
Connolly then issued her own address as Uachtarán na hEireann.
'I stand before you humbly and proudly' - President Catherine Connolly speaks of a 'new Republic' in inauguration speech. pic.twitter.com/ISPl2uAYBu
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In it, Connolly said she will ensure "all voices are represented, heard and valued".
Beginning her speech after signing her declaration of office, Connolly remarked: "We can breathe a little – and relax."
She said she stood before those gathered "humbly and proudly as the 10th president of this beautiful country".
Connolly said: "The people have spoken and have given their President a powerful mandate to articulate their vision for a new republic, a republic worthy of its name where everyone is valued and diversity is cherished, where sustainable solutions are urgently implemented, and where a home is a fundamental human right."

President Catherine Connolly.
The President added: "The change that led to this joyful day began with the small group of elected representatives and volunteers facing what appeared to be insurmountable challenges.
"We were led to believe that it was too great a leap, that our ideas were too far out, too left – at odds with the prevailing narrative.
"In shared conversations all over the country, however, it became evident that the dominant narrative did not reflect or represent people’s values and concerns.
"Time and time again, people spoke of how it served to silence, to other, to label, to exclude and to stifle critical thinking.
"Along with that meaningful engagement, we saw the emergence of hope. We saw the emergence of joy, along with the courage and determination of people to use their voices to shape a country that we can be proud of."
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Connolly continued by saying that Ireland faces the ‘inextricably linked’ threats of climate change and ongoing wars.
She stated: "Unfortunately, we have all become witnesses to ongoing wars and genocide.
"Given our history, the normalisation of war and genocide has never been and will never be acceptable to us.
"As a sovereign independent nation with the long and cherished tradition of neutrality and an uninterrupted record of peacekeeping since 1958, Ireland is particularly well-placed to lead and articulate alternative diplomatic solutions to conflict and war.
"Indeed, our experience of colonisation and resistance of a catastrophic man-made famine and forced immigration gives us a lived, understanding of dispossession, hunger and war and a mandate for Ireland to lead."
Connolly said Ireland could take "real pride in the success" of the Good Friday Agreement as a "model for the peaceful resolution of conflict".
She said she looked forward to her "first official visit to the north", as she referenced Article 3 of the constitution, which sets out the firm wish of the Irish people for a united Ireland under consent.
* This article was originally published on Extra.ie.
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