Tens of thousands of South Sudanese have watched the raising of the new country's flag at an independence ceremony in the capital, Juba.
Salva Kiir signed the constitution and took his oath of office in front of the jubilant crowds, becoming the president of the world's newest nation.
Sudan's President Omar al-Bashir and UN chief Ban Ki-moon are among dignitaries watching the historic events.
Sudan earlier became the first state to officially recognise its new neighbour.
The world's newest nation was born at midnight local South Sudanese time (2100 GMT), 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.
South Sudan became the 193rd country recognised by the UN and the 54th UN member state in Africa.
Celebrations in Juba began at midnight (2100 GMT). A countdown clock in the city centre reached zero and the new national anthem was played on television.
Saturday's independence ceremony is being held at the mausoleum of the late rebel leader John Garang, who died just months after signing the peace deal that ended Africa's longest-running conflict.
The BBC's Will Ross in Juba says people have flocked to the event on a baking hot day - some of them climbing trees to get a view.
There is a heartfelt display of excitement mixed with relief that after many years of feeling oppressed the tie with the north has finally been cut, he says.
The Speaker of the South Sudan Legislative Assembly, James Wani Igga, read out the Proclamation of the Independence.
Crowds then cheered as Sudan's national flag was lowered and the new flag of South Sudan was raised as trumpets played the new national anthem.
Afterwards the master of ceremonies told the crowd that President Kiir has decided not to hand over the flag of the Republic of Sudan.
"It shall be kept be kept in the archives of South Sudan in recognition of the common history that we have lived together," he told the crowd.
Our reporter says people in the crowds have been saying it is a moment to celebrate but they are also talking about the many lost relatives who died during the war.
Earlier, Mr Bashir arrived at Juba airport where he was greeted by Mr Kiir.
Other dignitaries attending the celebrations include former US Secretary of State Colin Powell and the US permanent representative to the UN, Susan Rice.
Under the Comprehensive Peace Agreement, a referendum was held on independence, which was approved by more than 99% of voters.
The new country is rich in oil, but one of the least developed countries in the world, where one in seven children dies before the age of five.
Unresolved disputes between the north and south, particularly over the new border, have also raised the possibility of renewed conflict.
President Bashir, who agreed the 2005 peace deal with the Sudan People's Liberation Army (SPLA), stressed his country's "readiness to work with our southern brothers and help them set up their state so that, God willing, this state will be stable and develop".
"The co-operation between us will be excellent, particularly when it comes to marking and preserving the border so there is a movement of citizens and goods via this border," he told journalists in Khartoum.
Fears of fresh violence resurfaced after recent fighting in two border areas, Abyei and South Kordofan, where some 170,000 people have been forced from their homes.
But separate deals - and the withdrawal of rival forces from the border - have calmed tensions.
The UN Security Council has passed a resolution approving a 7,000-strong peacekeeping force for South Sudan - but this is basically a rebranding of the force that was already in Sudan, mostly in the south.
Our correspondent says keeping both the north and the south stable long after the celebratory parties have ended will be a challenge.
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.
Analysts say the priority for Khartoum will be to negotiate a favourable deal on oil revenue, as most oilfields lie in the south. At present, the revenues are being shared equally.
Khartoum has some leverage, as most of the oil pipelines flow north to Port Sudan on the Red Sea.
Citizenship is also a key sticking point. A new law passed by the National Assembly in Khartoum has withdrawn Sudanese citizenship from all southerners.
The UN refugee agency has urged both governments to prevent statelessness.