The Irish cabinet has also moved the border counties of Donegal, Cavan and Monaghan to level four restrictions, the second highest level.
This will include closing gyms, pools and leisure centres, and non-essential retail and limiting weddings to six.
People will also be encouraged to work from home.
They will come into force from midnight on Thursday until 10 November.
Taoiseach Micheál Martin said the picture emerging in Donegal, Monaghan and Cavan was “very worrying”.
He said the national figure of cases per 100,000 in the last 14 days was 190.7, but in Monaghan it was 360, in Donegal it was 353 and in Cavan it was 571.
“The implications for local hospitals and public health in these areas is obvious and very serious, so too is the need for immediate escalation in restrictions in those three counties,” he said.
The new rules come after the introduction of tighter restrictions on schools and hospitality in Northern Ireland.
The Irish government is following a five-level system for dealing with coronavirus.
Movement has already been restricted between counties in the Republic.
Garda checkpoints have been set up across the country.
In the Republic of Ireland, a further 1,095 new coronavirus cases were confirmed on Wednesday, and five additional deaths.
There have now been a total of 45,243 confirmed cases of Covid-19 in the country and 1,835 deaths.
Ireland’s chief medical officer Dr Tony Holohan said it was “extremely concerning”.
He reiterated the need to reduce contact with other people.
“That means staying at home, working from home where possible, practising physical distancing and stopping discretionary socialising,” he said.
Tánaiste Leo Varadkar said “as best we can” there was an effort to co-ordinate and co-operate with Northern Ireland.
However, he added that Stormont had not agreed to an all-island approach.
The country’s coronavirus plan allows for schools to remain open even at level five.
He said that was based on international evidence which shows that schools, particularly primary schools, were not a major cause of transmission.
Mr Varadkar said the government would listen to what the National Public Health Emergency Team said, and to the new modelling on the positivity rate nationally, before making any further decisions.