The new name will be used both domestically and internationally, while Macedonia will also amend its constitution as part of the deal.
We have a good agreement that covers all of the conditions put forth by the Greek side.
Prime Minister Alexis Tsipras said Tuesday that a deal had been reached to give Macedonia a different name, addressing all of Greece's concerns.
The issue has been heated, and one with very real repercussions for Macedonia: It hasn't been able to join the European Union or North Atlantic Treaty Organisation because Greece opposed its name.
"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.
Ancient Macedonia was the cradle of Alexander the Great's empire, a point of pride to Greeks today.
"There is no way back", Zaev told a news conference after a telephone conversation with his Greek counterpart Alexis Tsipras.
"I now call on both countries to finalise the agreement reached by the two leaders". He added that a meeting between the two could take place.
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In Skopje, meanwhile, Macedonian president Gjorge Ivanov said earlier in the day that he remained opposed to writing the new name into the constitution, a move meant to show the change is permanent and binding for domestic and global use.
"The European Union perspective of the Western Balkans ... remains the most powerful stabilising force for the region", they added.
NATO Secretary General Jens Stoltenberg said Tuesday's "historic agreement" was "testament to many years of patient diplomacy", and called on the two countries' prime ministers to finalize the deal.
"This will set Skopje on its path to North Atlantic Treaty Organisation membership". "Thanks to you, the impossible is becoming possible", he tweeted.
The country will now be known as the Republic of North Macedonia, or Severna Makedonija in Macedonian.
Defense Minister Panos Kammenos, whose right-wing Independent Greeks party is Tsipras' governing coalition partner, said he would oppose an agreement in a parliamentary vote, meaning the left-wing prime minister will need to seek support from political opponents.
Because of Greek objections, Macedonia was admitted to the United Nations under a provisional name, the Former Yugoslav Republic of Macedonia. Tsipras and Zaev spoke on the phone yesterday for an hour, and said they would continue their conversation today.
Skopje also needs to revise its constitution, Tsipras said, before Greece ratifies the deal.
However, several critics said the deal is problematic in certain areas - citing gray areas and the extensive time frame which includes several obstacles.