The number of suspected swine flu cases in Scotland now stands at 32, the Scottish Government has confirmed.
Health Secretary Nicola Sturgeon said tests in 15 other cases had returned negative results for the virus.
She told MSPs the total number included 24 new suspected cases, all of them people with travel connections to Mexico or other affected areas.
Ms Sturgeon stressed the risk to the public was "low" and officials were working hard to keep it that way.
There have been three more confirmed cases in the UK - two adults, in London and Birmingham, and a girl in Devon.
All three of the new confirmed cases had recently travelled from Mexico.
Prime Minister Gordon Brown said they had mild symptoms and were responding well to treatment.
Health Secretary Nicola Sturgeon's statement to MSPs
Iain and Dawn Askham, a newly-wed couple from Polmont near Falkirk, were the first Britons to be diagnosed with swine flu.
They were admitted to a Lanarkshire hospital at the weekend after honeymooning in Mexico.
Ms Sturgeon said the Askhams were continuing to recover in hospital, adding that eight of the nine people who had been in close contact with the couple and had displayed mild symptoms had tested negative for swine flu.
Test results in the ninth case are still outstanding.
Suspected cases have now been investigated in the Forth Valley, Grampian, Lothian, Lanarkshire, Tayside, Greater Glasgow and Clyde, Ayrshire and Arran and Highland health board areas.
Ms Sturgeon said: "The test results suggest that, to date, as far as we know, we have managed to prevent the spread of infection within Scotland.
"That, at this stage, is encouraging."
Parliament was told that, of the 14 cases under consideration reported on Tuesday, one had now been identified as a close contact of the Askhams.
The rest were people who travelled from Mexico or other affected areas.
Ms Sturgeon said five of those cases had also tested negative for influenza, while two had been "declassified" as cases which were no longer suspect.
The results in seven of the cases - including the contact - remain outstanding.
The further 24 suspected cases - under investigation as of 1000 BST on Wednesday - surfaced in the Ayrshire and Arran, Greater Glasgow and Clyde, Grampian, Lanarkshire, Highland and Lothian health board areas.
NHS 24 has increased its staffing resource and the service is performing extremely well with the rise in call demand
Ms Sturgeon said the Scottish Government's top priority was to disrupt the spread of infection by assuming all suspected swine flu cases to be positive until proven otherwise.
She said the second priority was to learn as much as possible about the nature of the virus, adding: "Our extensive contact of tracing suspected cases, something that not all countries do as systematically as we do, puts us in a potentially strong position to learn more about the virus and how it behaves.
"That is, of course, vital in helping us shape our future response to it."
Scotland currently has a large enough stockpile of antiviral drugs to treat half the population if necessary, said the health secretary.
According to NHS 24, helpline centres were taking 23% more calls than usual, with people seeking reassurance and advice, while daily website visits had risen by 70%.
Medical director Dr George Crooks said the service received 3,200 phone calls on Tuesday, with 10% of the inquiries relating to swine flu.
"However, only a small number of these calls relate to people with flu-like symptoms who have returned from affected areas abroad and many of these symptoms are common problems often experienced by returning travellers," he said.
"NHS 24 has increased its staffing resource and the service is performing extremely well with the rise in call demand."
In the next few days, leaflets with information about swine flu will be sent to every household in the UK.
Organisations are being warned to make their own preparations for a possible pandemic.
Meanwhile, the global outbreak has spread further, with the first death outside Mexico confirmed in the United States.
US health officials revealed that a child had died of swine flu in Texas.
About 160 deaths in Mexico are being linked to the virus, though only a handful of those have been confirmed.
Samples of the new strain of swine flu found in Mexico and the US are being sent to Britain where work will start on analysing the virus.
Leading bacteriologist Professor Hugh Pennington said the test results from Scotland would help answer how transmissible the virus was from person to person.
The World Health Organisation has raised its alert level to four - two steps short of declaring a pandemic.
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