"The name [will be] the Republic of North Macedonia translated into all languages for general use", Zaev said, as aired by the Macedonian TV channels.
"Maybe what has the most historic gravity and value for Greece (is that) according to this accord. our northern neighbours don't have, and can not assert, any link to the ancient Greek culture of Macedonia", Tsipras said in a televised address on Tuesday evening.
Due to Greek opposition to its northern neighbor sharing a name with a Greek province, Macedonia has been recognized by the United Nations and other global organizations as the Former Yugoslav Republic of Macedonia.
Greek Prime Minister Alexis Tsipras said that he had reached a "good agreement" with Skopje on the constitutional name of the Republic of Macedonia, adding that it satisfied all of Athens' preconditions.
"There is no way back", Zaev told a news conference after a telephone conversation with his Greek counterpart Alexis Tsipras.
Greek Prime Minister Alexis Tsipras said the deal ends any claim he believes Macedonia may have had on Greek territory.
They also urged the Council to endorse opening EU accession talks with Macedonia, which the European Commission recommended in April.
'This historic agreement is testament to many years of patient diplomacy, and to the willingness of these two leaders to solve a dispute which has affected the region for too long, ' NATO Secretary General Jens Stoltenberg said in a statement.
Matthew Nimetz said in a statement Tuesday he has "no doubt this agreement will lead to a period of enhanced relations between the two neighboring countries and especially between their people".
GETTYThe tiny Balkan country has repeatedly tried to join both the EU and Nato
By solving the dispute which has lingered for more than 25 years, Macedonia will remove the biggest obstacle on its path towards European Union and North Atlantic Treaty Organisation accession.
"It will help to consolidate peace and stability across the wider Western Balkans", he said.
Zaev said he will put the deal to a vote in a referendum, while the Greek parliament will consider ratification before the end of the year.
In Skopje meanwhile, Macedonian President Gjorge Ivanov said earlier in the day that he remained opposed to a constitutional change that would likely be included in the draft deal, to provide an assurance that the name change was permanent and binding for domestic and global use.
He said Macedonia would revise its constitution for the name change and that the deal "secures the historic heritage of ancient Greek Macedonia".
Mr Tsipras said the deal met the concerns of Greece that a revised name for Macedonia be a "compound" name with a geographical qualifier.
'I am keeping my fingers crossed, ' he said.
In 1994 Greece imposed an economic embargo on Macedonia and prevented it from using the port at Thessaloniki, Skopje's main trading post. Both countries agreed that the new name would be used both internationally and bilaterally.
When Macedonia peacefully gained independence from Yugoslavia in 1991, Greece balked at the name, saying it implied territorial ambitions toward Greece.
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