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Police said they were responding to a report of a dispute at a house at Mullach Alainn near Omeath in County Louth at about 18:00 local time on Sunday when the incident occurred.
A woman was also seriously injured in the shooting.
The police officer who died was Anthony Golden, 36, a father of three.
BBC NI's Dublin Correspondent Shane Harrison said the gunman had been named locally as 24-year-old Adrian Crevan Mackin.
He was facing charges of membership of a dissident republican organisation and was out on bail.
Commissioner Noirín O'Sullivan, the head of the Irish police force, said the incident was a "terrible tragedy".
Dangers
"I am deeply saddened to hear of the death of my colleague in a fatal shooting incident this evening while carrying out their duties," she said.
"This incident highlights the unique nature of the job carried out by the men and women of [the police force], and the dangers they face every day."
Two men died and a woman was seriously injured in the shooting
Frances Fitzgerald, the Irish justice minister, said the police officer's death would be "mourned by the entire nation".
"The fact that [he] has laid down his life while protecting the community is a cause of great sadness," she said.
Perspective
"While no words at this time can be expected to console his wife and children, his family, his colleagues and all who loved him, they know that he gave his life protecting the community he was so proud to serve."
Irish President Michael D Higgins contacted Commissioner O'Sullivan to offer his condolences.
"I want to express my deepest sympathy to the family and friends of the Garda who has so tragically lost his life while responding to this incident," he said.
"My thoughts at this time are also with all others who have been affected by the events in Omeath earlier this evening."
Northern Ireland Justice Minister David Ford said he was "absolutely horrified" to hear about the deaths.
"I spoke to my colleague Frances Fitzgerald in Dublin to express my sympathy and asked her to pass that on to garda officers," he said.
The Garda Representative Association, which represents rank and file members of the Irish police force, said it was "shocked and saddened" by the officer's death.
Its president Dermot O'Brien said: "This dreadful, dreadful news is the deepest fear of every police family. This puts everything we do into perspective."
The scene of the shooting has been sealed off for a forensic examination.
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