Vaccines will be given to children aged 12 and over with underlying health conditions, as well as those in the same household as someone who is immunosuppressed.
Youngsters in the three months before their 18th birthday will also be included.
The majority of 12 to 17-year-olds will not be given a vaccine, though this is still under consideration by the Joint Committee on Vaccination and Immunisation (JCVI).
It follows confirmation by UK vaccines minister Nadhim Zahawi this will be the policy in England and elsewhere in the UK.
Children will be given the Pfizer vaccine, which is the only brand approved for use in 12 to 15 year olds by the Medicines and Healthcare products Regulatory Agency (MHRA).
Mr Zahawi insisted the decision not to give vaccines to all children was not because of limits to vaccine supply.
Vulnerable teenagers aged 16 or 17 have already been included in the existing vaccine rollout.
Dr Smith said: “In line with the JCVI’s most recent advice, children and young people aged 12 to 15 years old who have particular underlying conditions that put them at increased risk of serious consequences from Covid-19 will now be offered the Covid-19 vaccine.
“Children and young people aged 12 years and over who are household contacts of people who are immunosuppressed will also be offered the vaccine. As we have done throughout the pandemic, we will continue to follow the expert advice from the JCVI.
“Work is underway to deliver this in the forthcoming weeks.
“The JCVI will continue to fully assess all available data as a priority and advice will be kept under active review. I know that the committee will give any additional update to advice as soon as the evidence supports this and, in the meantime, we will offer the vaccine to those children and young people currently recommended.
“We are already vaccinating young people who will turn 18 by 31st July and we will continue to roll-out invites for those who will turn 18 years old within a three-month window.
“It remains crucial that everyone who is offered a vaccination takes up the offer.”
The Royal College of Paediatrics and Child Health (RCPCH) welcomed the news, but called urgently for more detail about which children would be eligible.
Contact, a charity for families with disabled children, called the news a “huge relief”, but said some families of children who had been shielding, but are not eligible for the vaccine may be concerned.
The RCPCH said: “We are very pleased that the government has now accepted the advice of this expert group (which contains a number of paediatricians) – advice which was made available to government some weeks ago.
“The JCVI advice reiterates what the evidence tells us – that most children are at minimal risk of being made seriously ill by Covid.
"Having looked at the available national and international data, the committee has weighed in the balance the benefit to children over 12 of being vaccinated, against the very small but important risk of potential side effects from the vaccine. They have decided that for children who are otherwise healthy, the risk is not outweighed by the benefit.
“The committee has, however, specified that some specific groups of children and young people aged 12 and over are at increased risk of hospitalisation from Covid and have recommended that these children be offered the Pfizer BioNTech vaccine, the only vaccine authorised by the MHRA for 12-17 year olds in the UK.
"Clarity and more detail about who exactly will be included in these groups is required as a matter of urgency both for families and also for paediatricians and primary care professionals who will inevitably be asked for advice from patients and their parents or carers.
“The JCVI has also noted that some children and young people, although not in these at-risk groups, may wish to have the vaccine in order to protect household members who are immune-suppressed. Provided they have proper information to make a decision, we believe children’s wishes to protect those with whom they live, should be listened to and respected.”
Professor Adam Finn, a professor of paediatrics at the University of Bristol, told a briefing this was an “evidence-based decision” that would be “constantly” kept under review.
He said: “The virus very rarely affects children seriously, particularly children who are healthy. The deaths and serious cases that we’ve seen in children have, for the most part, been in children with a number of underlying conditions which seem to predispose them to serious disease.
“In fact, those children are a much narrower group than the ones that we originally conceived might be at risk and designated clinically extremely vulnerable. It’s critical that informed consent is obtained in a careful way both from the parents and the child themselves.”
Amanda Batten, chief executive of Contact, said the announcement would be a “huge relief” to families of clinically vulnerable young people.
She said: "Some are still shielding and are desperate for their child to be vaccinated, so they can get back to school and friends. Since the Pfizer vaccine was found safe for 12-15 year olds by the UK regulator in June, they have felt abandoned and frustrated that the guidance has taken so long. Many families felt fearful as cases have risen and restrictions lifted.
“It’s vital that the roll-out to the identified groups is done as swiftly as possible."
She added: “We are concerned there will be some families disappointed because their child doesn’t fall into the categories identified today as able to get the vaccine.
"They were asked to shield at the advice of government and some are understandably still worried. They need reassurance that their child is no longer at risk of serious illness with data and facts used by the JCVI in drawing up their list.”