Leaders across the globe have been sending their congratulations to South Sudan on the day it became the world's newest nation.
Statements recognising South Sudan's nationhood flowed from the US, UK, Russia and others as tens of thousands watched an independence and flag-raising ceremony in the capital, Juba.
Salva Kiir took the oath as president.
Sudan's President Omar al-Bashir attended and called on the US to end sanctions against his country.
US President Barack Obama said in a statement he was "proud to declare that the United States formally recognises the Republic of South Sudan as a sovereign and independent state upon this day, July 9 2011".
He said: "A proud flag flies over Juba and the map of the world has been redrawn. These symbols speak to the blood that has been spilled, the tears that have been shed, the ballots that have been cast, and the hopes that have been realised by so many millions of people."
But he did not mention the long-standing sanctions against Khartoum. Washington still lists Sudan as a state sponsor of terrorism.
In his speech to the independence ceremony, Mr Bashir said: "We congratulate our brothers in the south for the establishment of their new state. We share their joy and celebration. The will of the people of the south has to be respected."
He added: "We call on US President Barack Obama to deliver on his commitment he announced to lift the unilateral sanctions on Sudan to open to way to normalise his country's relations with Sudan."
Mr Obama indicated more work needed to be done, particularly in the disputed border regions of Abyei and Southern Kordofan.
UK Prime Minister David Cameron issued a statement recognising South Sudan, saying: "This is an historic day, for South Sudan and the whole of Africa."
Russian President Dmitry Medvedev sent a telegram of congratulation and China's special envoy sent President Hu Jintao's "warmest congratulations".
South African President Jacob Zuma said: "We have always aspired to witness the dawn of peace, security and stability prevailing in the whole of the Sudan. That dream is coming to fruition."
'Hearts and minds'
The BBC's Will Ross watched the independence ceremony in Juba, saying the euphoria built as people waved goodbye to the north of Sudan and the flag of South Sudan was raised.
Wearing his trademark black cowboy hat, the South's new President Salva Kiir was sworn in and pledged better times ahead.
He told guests including UN Secretary General Ban Ki-moon, US ambassador to the UN Susan Rice and EU foreign policy chief Catherine Ashton: "Our martyrs did not die in vain... We have waited for more than 56 years for this day. It is a day that will be forever engraved on our hearts and minds."
The world's newest nation was born at midnight local South Sudanese time (2100 GMT Friday), the climax of a process made possible by the 2005 peace deal that ended a long civil war.
The South's independence follows decades of conflict with the north in which some 1.5 million people died.
Saturday's independence ceremony was held at the mausoleum of the late rebel leader John Garang, who died just months after signing the peace deal with Mr Bashir that ended Africa's longest-running conflict.
Meanwhile, in Khartoum, for most people it has been a low-key day, the BBC James Copnall reports from the northern capital.
However, scores of men gathered near the Blue Nile holding giant Sudanese flags and shouting: "Allahu Akbar [God is great]."
Under the Comprehensive Peace Agreement, a referendum was held on independence, which was approved by more than 99% of voters.
South Sudan is rich in oil, but is one of the least developed countries in the world, where one in seven children dies before the age of five.
Correspondents say keeping both the north and the south stable long after the celebratory parties have ended will be a challenge.
Fears of a new war resurfaced after recent fighting in Abyei and South Kordofan, where some 170,000 people have been forced from their homes.
Separate deals - and the withdrawal of rival forces from the border - have calmed tensions.
But the two sides must still decide on issues such as drawing up the new border and how to divide Sudan's debts and oil wealth.
Citizenship is also a key sticking point. A new law passed by the National Assembly in Khartoum has withdrawn Sudanese citizenship from all southerners.