Most people infected with the new strain - believed to be a hybrid of swine, bird and human influenza - have either travelled to Mexico or been close to someone who has.
US overtakes Mexico
In the United States, the Centre for Disease Control and Prevention reported a jump in the number of confirmed cases from 896 sufferers to 1,639 in 43 states, overtaking its southern neighbour Mexico, which put its confirmed cases at 1,319 on Friday.
The US centers said the increase had been expected after a backlog of laboratory tests was cleared.
"We do expect to see the numbers climbing," Dave Daigle, a spokesman for the centre, said.
Two people have died in the US after contracting H1N1 and officials expect that the virus will spread to all states.
However, Barack Obama, the US president, was cautiously optimistic that the outbreak would not be as bad as originally feared.
"We are seeing that the virus may not have been as virulent as we at first feared," he said at the White House.
"But we are not out of the woods yet. We still have to take precautions."
South of the US border, Panama became the latest Central American nation to confirm a case of the A(H1N1) virus while Guatemala said it had two people who had caught the disease.
Rosario Turner, the Panamanian health minister, said the unidentified youth, who arrived in the country on a flight from the United States, was in quarantine.
"He is receiving treatment and his condition is stable and does not require hospitalization," she said.
Brazil confirmed on Friday its first case of of the influenza A(H1N1) transmitted from person to person within the country.
Jose Gomes Temporao, the Brazilian health minister, said a new case, one of two found on Friday, was "up to now the only case of person-to-person transmission (of the virus) in Brazil."
The discovery brought to six the total number of cases of the virus detected in Latin America's largest nation.
Australia also reported on Saturday its first case of new flu strain, after a woman tested positive for the disease as she returned from a trip to the United States, the government said.
In Hong Kong, more than 280 guests and staff were finally allowed to leave a hotel after spending a week in quarantine.
The Mexican guest, who tested positive and had stayed at the hotel, was also released from hospital on Friday.
Elsewhere in Asia, Japan's health ministry said on Saturday a teacher and two students tested positive for H1N1 in the first confirmed cases of the virus in Japan.
The three, including a man in his 40s and two teenagers from Osaka, arrived at Tokyo's Narita international airport on Friday from Detroit, the ministry said.
"We confirmed the first case of the new type influenza in Japan," said Atsushi Kitamura, a health ministry spokesman.
They were then taken to hospital, where further tests were conducted, he said. Kyodo News said all three who tested positive were male.
Despite Friday's developments, the World Health Organisation (WHO) kept its alert level at five, indicating that a pandemic is "imminent" rather than actually taking place.
"We still remain in stage five. We have no evidence of community transmission," Sylvie Briand, the acting director of the global influenza programme at WHO, said.
Although the virus has been confirmed in 25 countries across the world, the vast majorities of cases have been in North America.