Schools in and around Mexico City have been closed until 6 May, and some 70% of bars and restaurants in the capital have been temporarily closed.
People are being strongly urged to avoid shaking hands, and the US embassy has advised visitors to the country to keep at least six feet (1.8m) from other people.
Mexico's Health Secretary, Jose Cordova, said a total of 1,324 people had been admitted to hospital with suspected symptoms since 13 April and were being tested for the virus.
"In that same period, 81 deaths were recorded probably linked to the virus but only in 20 cases we have the laboratory tests to confirm it," he said.
Mexico's President Felipe Calderon has announced emergency measures to deal with the situation.
1918: The Spanish flu pandemic remains the most devastating outbreak of modern times. Caused by a form of the H1N1 strain of flu, it is estimated that up to 40% of the world's population were infected, and more than 50 million people died, with young adults particularly badly affected
1957: Asian flu killed two million people around the world. Caused by a human form of the virus, H2N2, combining with a mutated strain found in wild ducks. The impact of the pandemic was minimised by rapid action by health authorities. The elderly were particularly vulnerable
1968: An outbreak first detected in Hong Kong, and caused by a strain known as H3N2, killed up to one million people globally, with those over 65 most likely to die
They include powers to isolate individuals suspected of having the virus without fear of legal repercussions.
In the US, seven people in California, two people in Texas, and two people in Kansas have been infected with the new strain.
In New York, city health commissioner Dr Thomas Frieden said preliminary tests conducted on the ailing students showed they were possible cases of swine flu.
Further tests will clarify if it was the same strain that was detected in the other three states.
Following a meeting of its emergency committee on Saturday, the WHO said the virus had the potential to become a pandemic but it was too early to say whether that would happen.
WHO Director General Margaret Chan said recent events constituted "a public health emergency of international concern" and that countries needed to co-operate in heightening surveillance.
Face masks are handed out, while the head of the WHO voices concern
The WHO is advising all countries to be vigilant for seasonally unusual flu or pneumonia-like symptoms among their populations - particularly among young healthy adults, a characteristic of past pandemics.
Officials said most of those killed so far in Mexico were young adults - rather than more vulnerable children and the elderly.
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