He said he hoped to contest the Louth constituency, near the Irish border.
He said his Stormont replacement would be chosen this week, while he would remain as MP for West Belfast until the Irish general election is called.
Mr Adams said the main reasons for his surprise decision were the economic crisis and the need for new ideas.
He said the current Fianna Fail/Green coalition government was "probably the most unpopular in the history of the state".
By Mark SimpsonBBC Ireland Correspondent
As the peace process has settled down in recent years, Sinn Fein president Gerry Adams has often looked bored.
There is no longer a new political crisis every week in Belfast. As a result, Mr Adams has a reduced workload and a much lower profile.
That may not be the main reason he has decided to run for the Irish parliament, but it may well be a factor.
While his right-hand man Martin McGuinness has, as deputy first minister, become the face of Sinn Fein at Stormont, Mr Adams has had a peripheral role.
He spends a lot of time south of the border, but as an unelected politician he is effectively shouting from the sidelines.
At 62, he has decided to try to re-invent himself. If elected he would try to lead a surge in support for Sinn Fein in the south.
If he loses, it could spell the end of his long political career.
"As leader of the only all-Ireland party with an all-island mandate, I have a choice to make whether to stay in west Belfast, a place that I love, or to seek a mandate in another constituency in the south.
"West Belfast is my home. It is where Colette and our family are and where I live.
"But after thoughtful consideration, and with the support of colleagues, I have decided to put my name forward for Louth. If elected for this constituency I will work and stay here and travel home when possible."
The Irish prime minister, Brian Cowen, must call a general election before July 2012 and it is thought the election may happen next year.
Louth is currently represented by Sinn Fein's Arthur Morgan, who has said he plans to step down at the next election.
Sinn Fein has five MPs, including Mr Adams, and four TDs in the Irish parliament.
Mr Adams made the announcement ahead of his keynote address at the Edentubber Commemoration in County Louth.
Sinn Fein deputy first minister Martin McGuinness said: "The decision by Gerry to leave one of the safest seats in Ireland to seek election to the Dail in Louth and to play a central role in the battle for Ireland's economic recovery is leadership in action."
Sinn Fein president Gerry Adams is to step down as an MP and Stormont assembly member to stand for election in the Irish Republic.