Taoiseach Micheál Martin has said the Government will take a cautious approach to international travel and non-essential travel is not advised until at least 20 July.
On that date, a so-called "green list" of countries with similar levels of the virus to Ireland will be published.
That will be followed by an easing of travel restrictions between those countries and Ireland.
Mr Martin said Covid-19 remains volatile judging by what is happening in other countries and Ireland needs to keep it suppressed.
He said that the Government's priorities include the reopening of schools as fully as possible in September along with increasing capacity in the health service to provide non-Covid treatments, and it will not do anything to jeopardise these aims.
Mr Martin said last weekend was a challenging one and rules around pubs must be strictly applied.
He said the Government was very worried about what had happened and nothing can be guaranteed when it comes to the reopening of all pubs planned for 20 July.
Countries on the so-called "green list" would have a disease level similar to Ireland, the Taoiseach added.
The list will be reviewed every fortnight.
Mr Martin said sectoral protocols will be published by the Department of Transport, and a list of exemptions will continue to apply to facilitate the transit of essential workers.
He said the current paper-based passenger locator form will be replaced by an electronic system to capture passenger data directly from airlines and ferries.
The regulations underpinning the passenger locator form will be extended until 20 July.
The Taoiseach said that "we know from reports seen at the weekend of an internationally volatile situation", and referenced Spain and the UK.
He said the situations in these countries are "informing" the Government's view, "as we see second waves emerging", he added.
"We have to make sure that in this country that the number of cases remains low, that community transmission remains low and that the R rate is kept below 1."
Mr Martin said the Department of Health will review testing at points of entry to the country.
He said there will be significantly heightened visibility of public health at border control, and gardaí at airports and ports.
Taoiseach Micheál Martin says the Department of Health will examine proposals for testing at points of entry into the country, adding there will be greater visibility of gardaí and health officials at ports and airports | More: https://t.co/p0oVRsHprq pic.twitter.com/rkPaHMXAFX— RTÉ News (@rtenews) July 6, 2020
There will also be a comprehensive public communications campaign.
Minister for Foreign Affairs Simon Coveney said public health officials believe that foreign travel is the number one risk for a second wave of the disease.
He said: "What we are going to do between now and 20 July is to put a green list together of countries at a similar risk or less than Ireland."
He said people travelling to or from those countries will not be asked to self-quarantine but for all other countries they will.
Minister Coveney said the requirement for passengers arriving into Ireland to self-quarantine is not legally mandatory.
He said the Government is not happy that the current passenger locator form is getting the required response.
Mr Coveney said they are working with airlines to move away from a paper-based form to a digital system and they will increase the capacity for follow-up checks as passenger numbers increase in the coming weeks.
He said if people give misinformation on their forms, then it is against the law.
Covid tracker app to be launched tomorrow
Minister for Health Stephen Donnelly has said the public health advice is against all non-essential overseas travel.
He said there are a lot of people who would like to see loved ones abroad but public health officials are very worried about the potential for a second surge brought about by an upswing in foreign travel.
Minister for Health Stephen Donnelly says the vast majority of people are complying with the public health advice against all non-essential foreign travel | More: https://t.co/p0oVRsHprq pic.twitter.com/FORBSXQYIR— RTÉ News (@rtenews) July 6, 2020
He said what is of most concern is that people do not travel to countries not on the green list, when it is published.
He said a Covid tracker app will be launched tomorrow that will be a "really powerful part" in tracking the disease. He said the app will allow people to control their own data, saying it is a very powerful tool and they encourage as many people as possible to download it.
Minister for Health Stephen Donnelly says the government will launch a Covid-19 contact tracing app tomorrow, which he says will play a powerful part in fighting the virus | More: https://t.co/p0oVRsHprq pic.twitter.com/BJXMYq6Ro3— RTÉ News (@rtenews) July 6, 2020
Minister Donnelly has said that the new app will not tell you whether you have tested positive, "you will be informed through the existing testing regime", he said.
He explained that if you have tested positive, and have downloaded the app, you then give permission to notify those who you have been in close contact with, that you have tested positive.
He added that these people can then avail of testing too.
"It's exciting; it means we can move quicker to a much more comprehensive 'seek and destroy' regime essentially. But we are all going to have to play our part. Download it, talk to our family, and encourage people to download it."