The Trump campaign has acknowledged in a statement that President Obama was born in the US.
The Republican candidate had been a leader of the "birther" movement that questioned Hawaii-born Mr Obama's citizenship.
But his campaign now accuses his Democratic rival Hillary Clinton of introducing the "smear" during the 2008 Democratic nomination contest.
There is no evidence to link Mrs Clinton to the birthers.
In reaction she tweeted that President Obama's successor "cannot and will not be the man who led the racist birther movement".
Does it matter where a country's leader is born?
The BBC's North America Reporter Anthony Zurcher says the statement signed by senior Trump advisor Jason Miller is far from an admission of error.
Instead, he says, Mr Miller laid the genesis of the birther rumours wrongfully at the feet of Hillary Clinton and her 2008 presidential campaign team.
When they raised questions, Mr Miller said, it was "vicious and conniving" behaviour. By broaching the topic three years later, Mr Trump had done a "great service" to the public and president, Mr Miller said.
The statement follows an interview with the Washington Post newspaper in which Mr Trump had declined to say Mr Obama had been born in the US, saying instead that he did not want to answer the question.
What is the origin of the 'birther' claim?
The claim is a conspiracy theory that Mr Obama was actually born in Kenya and is therefore ineligible to be president.
Reports in various US publications suggest it was circulated in 2008 by die-hard supporters of Mrs Clinton as it became clear that she was not going to win the Democratic nomination.
However there is no evidence that Mrs Clinton or her then campaign had anything to do with it.
The claim enjoyed a revival with some supporters of Republican candidate John McCain as he fell behind Mr Obama in polls, the Fact Check website reported.
When did Mr Trump get involved?
The billionaire became a vocal questioner of Mr Obama's citizenship during the run-up to his eventual re-election in 2012.
In April 2011, Mr Trump challenged Mr Obama to show his birth certificate, gaining approval from Republicans including former Alaska governor Sarah Palin.
The 2012 Republican candidate Mitt Romney referred to the discredited theory at a campaign rally in August that year, joking that no-one had asked to see his birth certificate - drawing swift condemnation from the Obama campaign.