Covid-19 Response Minister Chris Hipkins said the first New Zealanders have been fully vaccinated while the number of people who have refused the vaccine was around the "10 or 20" people.
There are no new community cases of Covid-19 but three in managed isolation and more than 6000 tests processed yesterday.
Hipkins asked people to keep any eye out for Covid-19 symptoms and to get tested if anyone has any.
"We can't be too careful."
More than 27,000 people have received the vaccine so far.
"We're making good progress on group 1," Hipkins said.
More than 1100 vaccinators have finished their training so far and more than 5000 people have registered their interest to become qualified to be part of the vaccine programme.
"It's a good, solid start."
He said the Government is working on a sustainable delivery programme.
All New Zealanders are asked to get vaccinated, Hipkins said.
Bliss has received both his vaccines and many frontline workers have received two shots now.
Bliss thanked the frontline workers for getting the vaccine, which adds "another layer of protection" to New Zealand.
New Zealand is now three and a half weeks into the programme and so far 90 per cent of frontline workers have got the vaccine.
He said 9600 NZDF workers will get the vaccine - 1600 have got it so far.
It will take six weeks to complete the vaccine rollout to the NZDF, he said.
Hipkins said the Government's target is getting "as many people vaccinated as possible" and ensuring people were coming forward to receive the jab.
"There will be a lot of demand at the end of the year," he said.
He said the Government will be administering "more than" 20,000 vaccines a day when the general rollout comes.
Hipkins said there will be "unders and overs" when it comes to the daily numbers for vaccines administered.
He said the Government has a good allocation until June, and from July the Government is expecting a lot more vaccines.
The Government will be doing "week by week" monitoring of how many people it has vaccinated.
On publicity campaigns, Hipkins said they started this weekend and will increase in the coming weeks and months.
The Government was looking at "role models" to take the vaccine and talk about its benefits publicly - this could include the All Blacks.
He confirmed these people wouldn't be paid to do this.
He said GPs will be given the most up-to-date information so they can help counter any misinformation.
Hipkins said the Government will have the ability to move vaccinators around the country to hotspots if that is needed.
He said the number of people saying no to a vaccine so far is "very small".
He said that number was in the "early double digits" - but he couldn't say exactly how many.
Hipkins said the frontlines were a "challenging" place to work, but he thanked those working in MIQ for their work.
He said some are feeling a bit of stigma around where they work - he called on New Zealanders to thank them for their work.
Hipkins said no sportspeople have been given the vaccine yet - but he is taking a Cabinet paper on this soon.
Brigadier Bliss said getting the vaccine was like any other vaccine he has ever received.
Hipkins said again that vaccination was not a requirement for all New Zealand, but said it is mandatory if someone wants to work on the frontlines.
Hipkins said the Lion King production team was not given preferential treatment and they acquired their own MIQ vouchers.
Hipkins said the Pullman Hotel is a "good facility" which has performed very well throughout the MIQ.
He said there has been work done to make it a lot more Covid-19 airtight.
The 1pm briefing comes after Bloomfield briefed MPs at the health select committee this morning.
He said he was "confident and can reassure New Zealanders" that the workforce to administer the Covid vaccine rollout will be ready.
The Government's vaccine safety campaign, he said, had a "soft launch" this weekend and would be ramped up soon.
He added that the Ministry of Health would "absolutely" approach the likes of All Blacks and other sports stars to help with the campaign.
The campaign would be focused on using "trusted" members of the public who can talk up the science and safety of the vaccines.
This comes as the Act Party continues to put pressure on the Government over its vaccination campaign.
Leader David Seymour pointed out that figures from Bloomberg show that so far, New Zealand has so far administered 18,000 Covid-19 vaccinations.
Australia, on the other hand, is now up to more than 161,800.
"The difference in the two vaccine rollouts is one country was ready, with a plan, and one country wasn't. It's that simple," he said.
"New Zealanders deserve better."
Meanwhile Nikki Turner, the director of the Immunisation Advisory Centre, briefed MPs on the Pfizer vaccine this morning.
Turner identified the logistics of the rollout as one of the big challenges for New Zealand, including moves to ensure people felt it was safe.
She said she did not believe concern about the vaccine was as widespread as many believed – and said the term "vaccine hesitancy" was not very useful, saying different people needed different information to make up their minds.
She did not want to set a target for herd immunity – rather, New Zealand should be striving for the highest level possible.
"There is unlikely to be a magic number."
But she warned that Covid-19 would keep coming into New Zealand, adding that the vaccine was not "a magic bullet".
Turner said "many more thousands" of people are ready to be trained to give Covid vaccinations.
There were two new cases of Covid-19 in managed isolation yesterday.
And, again, there were no new cases of Covid-19 in the community.