People must be "optimistic but patient" about the coronavirus situation in the UK and the end to restrictions, Boris Johnson has said.
The PM hailed the vaccine rollout, but warned that now was not the time to "relax".
He said there were still more people in hospital than at the peak of the first wave, and steps taken to ease lockdown should be "cautious but irreversible".
He is to set out a roadmap next week for lifting England's restrictions.
The government said on Sunday that it had met its pledge to offer at least one Covid vaccine dose to everyone in the top four priority groups in the UK. More than 15 million people have received a first dose, according to the latest government figures.
Speaking at Monday's Downing Street briefing, Mr Johnson hailed the "unprecedented national achievement" of hitting the target, but warned the threat from the virus remained "very real".
The prime minister warned that the government did not have all the "hard facts" about how the vaccine impacted infections and that there were still more people in hospital with Covid-19 than in April - with admissions running at 1,600 a day across the UK.
"We have to keep our foot to the floor," he said.
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Plan to end lockdown 'cautious but irreversible'
Mr Johnson said no decisions were being taken before he unveiled his roadmap for ending England's restrictions, but that the government wanted this lockdown "to be the last", with "cautious but irreversible" progress.
He added his plan next week would set out "as much as we possibly can about the route to normality, even though some things are very uncertain".
Although he said he was "increasingly optimistic" he admitted he could not give "an absolute cast-iron guarantee that we won't face further difficulties".
"I don't want people to think I'm not optimistic. There's been a big change - science is now in the ascendency over the disease," he added.
Mr Johnson also said the government was looking at introducing rapid lateral flow testing for venues such as nightclubs and some theatres that were unable to open last year.
"I think that, in combination with vaccination, will probably be the route forward," he said.
Michael Kill, chief executive of the Night Time Industries Association, said the "acknowledgement" would give late night businesses "the opportunity to deliver pilot schemes which will substantiate the ability for this sector to open safely at the appropriate time".
"But more importantly we will be able to plan and prepare for that opportunity," he added.
Could this be the last lockdown?
That is, of course, a huge shared hope and the government's stated aim.
It will soon be exactly a year since the first lockdown was brought in.
Who knew then that we'd be in a third period of national restrictions 12 months later?
Boris Johnson has been accused, at times, of over-promising and under-delivering during this crisis.
Today, again, he's urging caution - even as case rates fall and the vaccine rollout goes well.
That's because ministers really don't want to lift restrictions only to reimpose them again on a weary nation.
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It comes after more than 60 Conservative MPs wrote to Mr Johnson over the weekend, calling for a commitment to a "free life" and the ending of lockdown measures before May.
Steve Baker, the deputy chair of the lockdown-sceptic Covid Research Group (CRG) of Tory MPs leading the call, said schools should return on 8 March, hospitality should reopen by Easter and all other elements should be back to normal by 1 May - when all people in the top nine priority groups have been offered a vaccine.
Earlier, No 10 said the intention was to "start getting" pupils back into school on 8 March, but was not ruling out the possibility that they could be sent back in stages.
The PM's official spokesman said it was "important" the planned review of restrictions was carried out this week, with details to be set out in Mr Johnson's road map on Monday.
The spokesman added that he would not "pre-empt" what the road map might contain, but added: "it remains our priority to get kids back to school and we always said we want that to start from 8 March, if the science and evidence allows".
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The government is hoping to offer a jab to everyone in the first nine priority groups - including everyone over the age of 50 - by the end of April, Mr Johnson said.
This would run alongside giving second doses within the 12-week window to those in the top four priority groups, who have already had a first dose.
All those aged between 16 and 64 with underlying health conditions, as well as adult carers, are being contacted with the offer of a vaccination.
With every day that passes, it becomes clearer and clearer that the UK is winning its fight against Covid.
Science, as Boris Johnson said, is in the ascendancy.
It is easy to forget this virus only emerged just over a year ago.
But now the UK has approved three vaccines for use - two of which have already started to be rolled out at speed. Another two are being assessed by the regulator after promising trial results.
There is confidence the mutations, while presenting a challenge, can be overcome. Plans are already being made to update the vaccines if needed to work better against new variants.
No vaccine is 100% effective and some people may choose not to take it. That's why advances in treatment are vital, too.
A cheap steroid, dexamethasone, is being widely used to improve the survival rates of the sickest patients. And last week it emerged an arthritis drug, tocilizumab, could also further improve survival chances.
There is still a long way to go, but there is a cautious belief in both Westminster and among scientists that each step made from now on should be forward not back.
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Speaking at the same Downing Street briefing, NHS England's chief executive Sir Simon Stevens described the vaccine rollout as "two sprints and a marathon".
He said the UK had passed the "finishing line of the first sprint" and now had a "second sprint between now and 30 April" to extend the vaccine rollout to higher risk groups.
The "marathon" will involve vaccinating everybody in the country who is eligible, alongside the winter flu campaign and further booster shots if needed, Sir Simon added.
Meanwhile, on Monday, the UK recorded another 9,765 new coronavirus infections, according to the latest government figures. It is the first time the daily cases figure has been under 10,000 since 2 October.
A further 230 deaths within 28 days of a positive test were also recorded. Fewer deaths tend to be reported on Mondays due to a reporting lag over the weekend.
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