Every adult will have been offered the coronavirus vaccine by autumn, Health Secretary Matt Hancock has said.
The cabinet minister made the announcement as he told Sky News more than 200,000 people are currently getting a COVID-19 jab every day and the government is on course to reach its target of two million vaccinations a week.
But speaking on Sophy Ridge On Sunday, the health secretary said the current pressure on the NHS was "very bad" and repeated the need for people to follow lockdown rules and stay at home, warning "we can't let up" just because coronavirus injections are being rolled out.
This echoed the stark message by England's chief medical officer Chris Whitty that the NHS faced the "most dangerous situation" in living memory.
The health secretary told the BBC's Andrew Marr that "absolutely we're going to offer the vaccine to everybody" by autumn.
"We've got over 350 million doses on order," he said, "they're not all here yet, and we're rolling them out as fast as they get delivered."
Mr Hancock also spoke about enforcement of lockdown rules, saying he "absolutely" backed the police in strictly applying the law when it comes to COVID-19 restrictions, after two women were fined for driving five miles for a country walk, warning that "every flex can be fatal".
He told Ridge the government was set to meet its goal of vaccinating more than 13 million of the over-70s and the most vulnerable by mid-February - a target that requires two million jabs to be carried out a week.
Mr Hancock said: "Yes we're on course. The rate limiting factor at the moment is supply but that's increasing.
"I'm very glad to say that at the moment we're running at over 200,000 people being vaccinated every day.
"We've now vaccinated around one third of the over-80s in this country so we're making significant progress but there's still further expansion to go.
"This week we're opening mass vaccination centres. Big sites for instance at Epsom racecourse, there's seven going live this week with more to come next week where we will get through very large numbers of people."
Refusing to criticise the police in the face of complaints that some forces had been over-zealous in handing out fines, Mr Hancock stressed the need for people to stick to the rules.
He told Ridge: "The pressure on the NHS is very bad and so we need to bring the case rate down.
"As far as we can tell the vast majority of people are following the rules.
"Just because the vaccine's coming we can't let up, because the pressure on the NHS is right here right now."
Mr Hancock added: "I'm absolutely going to back the police because the challenge here is that every flex can be fatal.
"You might look at the rules and think, 'Well, it doesn't matter too much if I just do this or do that'.
"But these rules are not there as boundaries to be pushed, they are the limit to what people should be doing.
"The police are right to take very seriously the rules we have brought in. We haven't brought them in because we wanted to, we've brought them in because we had to."
The comments come after Home Secretary Priti Patel also gave her backing to police arguing there was "a need for strong enforcement where people are clearly breaking these rules" and pledged officers would "not hesitate to take action".
Labour leader Sir Keir Starmer stressed the importance of the "stay at home" message.
He said: "We are at the darkest stage of this pandemic now, it's a race between the virus and the vaccine and everybody needs to follow that government guidance."
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Meanwhile, Professor Adam Finn, a member of the Joint Committee on Vaccination and Immunisation (JCVI) said the vaccine rollout will already have prevented thousands of people from having to be admitted to hospital with the virus.
The professor of paediatrics at the University of Bristol told Ridge: "It's really too soon for the vaccine to have started having a measurable impact, but we can predict that already it's preventing cases just simply from the numbers of people who've received the vaccine which is now approaching one-and-a-half million people, and the rate of infection that's occurring which is really very high now.
"So there are certainly thousands of people already who have not been admitted to hospital and who will not be dying of this infection as a consequence of the programme that's begun in December."