Polish President Bronislaw Komorowski has conceded the election to conservative challenger Andrzej Duda following the release of exit polls.
They suggested Mr Duda had taken the run-off vote by 53% to 47%.
Mr Duda had edged Mr Komorowski, who had been the favourite, in the first round but did not gain the 50% needed to win outright.
The president has limited powers, but is head of the armed forces and can veto new laws.
The exit polls had been delayed after a woman died at a polling station on Sunday.
Official results are expected later on Monday.
"I respect your choice," Mr Komorowski told voters at a gathering of his supporters. "I wish my challenger a successful presidency."
Speaking to supporters in Warsaw, Mr Duda said: "Thank you President Bronislaw Komorowski for the rivalry of this presidential campaign and for your congratulations.
"Those who voted for me voted for change. Together we can change Poland."
The victory will be a wake-up call to Prime Minister Ewa Kopacz, an ally of Mr Komorowski, ahead of parliamentary elections this autumn.
Mr Komorowski, 62, took office five years ago after his predecessor, Lech Kaczynski, died in a plane crash.
Mr Komorowski had been the favourite according to previous opinion polls and had been looking for a second term.
Mr Duda, 43, is from the right-wing opposition Law and Justice party, which is led by former President Kaczynski's twin brother, Jaroslaw.
In the first round Mr Duda attracted most support in the more conservative eastern regions of the country, near the border with Ukraine and Belarus.
Credit: The BBC