Hallgrimsson was the surprise choice to replace Ireland interim manager John O'Shea, who has been in charge since Stephen Kenny's departure in November last year.
The unheralded 57-year-old will take charge of his first Ireland match when they face England in the Nations League in September.
"Ireland is rightly a proud footballing nation which has consistently produced talented players and enjoyed many memorable moments at major international tournaments," Hallgrimsson said.
"We have a young and exciting team that has genuine potential. I am looking forward to working closely with the players to help coach and guide them towards improved performances and results, ensuring we qualify and compete at major tournaments on a regular basis."
Hallgrimsson, who was joint Iceland coach with Lars Lagerback when they knocked England out of Euro 2016 in the last 16, has set his sights on leading Ireland to the 2026 World Cup.
"We have important and exciting games coming up in the UEFA Nations League later this year and a massive FIFA World Cup qualification campaign coming up next year," he said.
"It is my responsibility to coach, prepare and develop our team as best as possible to deliver results on the pitch and to make our supporters happy and proud."
Hallgrimsson resigned as Jamaica coach following their exit from the Copa America in June.
He had been in charge of Jamaica since 2022, catching the eye of FAI director of football Marc Canham.
"Earlier this year, we identified Heimir as our number one candidate whose capabilities and experience aligned with our criteria," Canham said.
"Not only does Heimir have significant experience at international level with two different countries, but crucially he also has a track record of qualifying for major international tournaments and taking teams up the FIFA world rankings."
