As of Friday, more than 19.6 million people had received their first dose - an increase of more than half a million in a day.
Meanwhile, the number of people who have had their second jab has risen to 768,810.
The government is aiming to offer all adults the vaccine by the end of July.
The next target is to offer a first dose to all over-50s by 15 April, as well as people aged 16-64 with certain underlying health conditions and unpaid carers for disabled and elderly people.
After that, they will start offering jabs to people by age group.
There are no plans yet to vaccinate children, although trials have been announced to test the Oxford University jab on children.
The latest government statistics show a total of 20,450,858 jabs have been administered in the UK - including 19,682,048 first doses, equivalent to more than one in three adults in the UK.
This includes more than 16.6 million first doses in England, more than 1.5 million in Scotland, 916,336 in Wales and 515,678 in Northern Ireland.
The figures also show another 7,434 cases and a further 290 people have died within 28 days of testing positive for coronavirus.
In a tweet on Saturday, Health Secretary Matt Hancock said he was "delighted" at the latest figures.
"The vaccine roll-out shows what we can achieve when we work together," he said.
Meanwhile, Labour is calling on the government to set out a "proper plan" for how the local elections in May will be run safely.
Earlier this month it was announced the elections would go ahead on 6 May but with some changes - for example plastic screens in polling stations and people told to bring their own pens.
But deputy leader Angela Rayner accused the government of a "failure to take any action to encourage people to sign up to vote safely from home", meaning people could be disinclined to vote or face "dangerous crowds at polling stations".
A Cabinet Office spokesman said a "strong set of measures" had been put in place to make polling stations Covid-secure and a comprehensive public information campaign would be launched soon "so that voters are fully informed about how to participate".