The first minister made her comments in response to the Sinn Fein president’s address at his party’s annual conference in Dublin on Sunday night.
An ard fheis motion was passed calling on the new government to produce a green paper on Irish unity, and to begin planning for unification.
Sinn Fein had organised its annual gathering to coincide with a number of events commemorating the centenary of the April 24, 1916 Easter Rising.
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Mrs Foster said: “One journalist summarised Gerry Adams’ conference speech by saying ‘the batteries are flat’. The reality is that Gerry Adams’ project has failed. It has run out of steam. More and more people see through it.
“All Gerry Adams has to offer is rhetoric. In 2016 more people than ever support Northern Ireland’s place in the Union. People can see that it makes sense politically, economically and socially.”
The DUP leader described her Sinn Fein counterpart’s plea on a government push for Irish unity as “unwise and a step backwards”.
“It would be wholly unacceptable for any new Irish government to give rise to such a campaign. As First Minister, my vision for Northern Ireland is within the United Kingdom and I will always work to strengthen our place in the Union.”
Referring to Mr Adams’ comments on the past, Mrs Foster said they were “insulting to those of us who were victims” of the Provisional IRA.
Mrs Foster continued,
“Gerry Adams stated: ‘We are proud of the men and women of 1916... we are equally proud of the men and women of the H Blocks and Armagh and of the 1981 hunger strikers and of the patriot dead from our time.’ This is a clear reference to the PIRA.
“In an attempt to shore up the base Sinn Fein ditched any concern for innocent victims. Is Sinn Fein proud of the cowards who planted the Enniskillen bomb, the La Mon bomb or who masterminded the disappeared?
“Gerry Adams is very clever at selectively quoting the past but he can rest assured that I will not allow him or anyone else in Sinn Fein to rewrite the past.”
As thousands turned out across the Republic to mark the centenary, Irish president Michael D Higgins led an inter-faith commemoration at Arbour Hill in Dublin.
A wreath was laid in honour of the rebel leaders in the cemetery where 14 of them are buried.
Relatives of the 16 leaders executed soon after the failed rebellion attended the event.
Archbishop of Dublin Diarmuid Martin spoke at a service in the Church of the Sacred Heart and urged people to seize the ideals of the rebels’ Proclamation.
He said the most striking thing from the accounts of priests in Arbour Hill in 1916 was the lack of rancour and revenge from the condemned men.
“We owe it to those leaders never to allow ourselves to resort to revenge and hatred. They fought for peace and reconciliation and justice in our country and for all who belong here or who come to our shores,” he said.
A number commemoration events also took place in Northern Ireland, including republican parades in Belfast.
Loyalists held a protest in Royal Avenue as a parade organised by a north Belfast grouping made its way through the city before making its way to Milltown cemetery.
There was a large police presence in the area with PSNI Land Rovers and crash barriers blocking off roads.
Protesters and marchers exchanged insults but there were no reports of any violence.
The protest had been organised by the Loyal People’s Protest and the Protestant Coalition.
Police welcomed the peaceful outcome of Sunday’s events in Belfast.
Following the conclusion of Sunday’s parades and associated protests, PSNI superintendent Darrin Jones said: “The aim of today’s police operation was to ensure the safety of everyone involved. We had an appropriate and proportionate operation in place and I am very pleased with how the day has passed.”
Supt Jones added: “The efforts made by community representatives and event organisers showed a real willingness to ensure all events passed off peacefully.”
Last month, hundreds of thousands of people lined the streets of Dublin on Easter Sunday as a major parade marked the culmination of a series of Irish Government-organised centenary commemorations.
The military parade - watched by an estimated crowd of 250,000, was the largest ever staged in the Republic.
An Orange Order parade in support of Belfast Orange Widows also took place in Belfast on Sunday.
The Ballynafeigh District parade, involving 100 participants and one band, made its way from the Ormeau Road to the Shankill Road via the city centre.
North Belfast DUP representative William Humphrey said he believed an overnight paint bomb attack on Clifton Street Orange Hall was a “clear attempt to raise tensions” ahead of the Orange Widows’ parade.