The new name - which in Macedonian is Severna Makedonija - will be used both domestically and internationally, while Macedonia will also amend its constitution as part of the deal.
Other Greeks said the new name should not even include the word Macedonia, while backers reject nationalism and said the dispute has gone on long enough.
'We have reached an accord, a good accord that covers all the conditions set by Greece, ' Tsipras said in televised comments after briefing Greece's president Prokopis Pavlopoulos on the accord.
He added that Macedonia would take a new "compound name with a geographical denominator", without spelling out the name.
In a statement, the Party of European Socialists (PES) welcomed the agreement reached by the two Prime Ministers.
Veteran envoy to Macedonia name dispute applauds
In Skopje, the opposition party VMRO-DPMNE, said Macedonian Prime Minister Zoran Zaev had informed party leader Hristijan Mickoski that he had "achieved a solution with Greece".
The doors to further Western expansion in the Balkans have been thrown wide open by a seeming breakthrough on the Macedonia name dispute.
Athens officials had objected to the name Macedonia, fearing territorial claims on a northern Greek province of the same name.
FYR Macedonia and Greece call it truce on the 27 year old ongoing name dispute.
Macedonia hopes that resolving the name dispute will help clear the way for it to join the European Union and North Atlantic Treaty Organisation.
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The US had sided with South Korea in the conflict. "But the President said he expected immediate action as a result of the talks". He said he would discuss it "at great length" at an upcoming news conference, and indicated it would be made available publicly.
"This will set Skopje on its path to North Atlantic Treaty Organisation membership".
"We have been solving a two-and-a-half-decade dispute. that has been drowning the country", he said, adding that the deal "will strengthen the Macedonian identity".
"I am keeping my fingers crossed", he said.
However, despite the what it seems an obvious sign of relief for both sides, the name resolution still has to be approved by the Macedonian people and the Greek parliament.
Greek media previously reported today that Zaev is going to talk with Tsipras and confirm the agreement that they accomplished earlier today.
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After about 30 strides, Mannino catches the child, according to the dashcam footage. The child was out for only a matter of minutes before Mannino spotted him.
Ancient Macedonia was the cradle of Alexander the Great's empire, a point of pride to Greeks today.
It will stress that the Macedonian language belongs to the South Slavic group of languages, Tsipras said.
"We wholeheartedly congratulate Prime Ministers Alexis Tsipras and Zoran Zaev for their determination and leadership in reaching this historic agreement", said EU foreign affairs chief Federica Mogherini and European Commissioner for Enlargement Negotiations Johannes Hahn in a joint statement.
But the breakthrough is now likely to see European Union leaders open accession talks with Macedonia at a summit in June and North Atlantic Treaty Organisation invite Macedonia to join the military alliance at a summit in July.
In 1994 Greece imposed an economic embargo on Macedonia and prevented it from using the port at Thessaloniki, Skopje's main trading post.
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Ever since the Republic of Macedonia declared its independence in 1991, Greece has been fighting the country over its name.