And a number of No campaigners have indicated that a court challenge to the result will now be brought.
The miserable turnout of only a third of voters has alarmed senior Government figures who will now decide whether to abandon future plans for Saturday voting.
The campaign will top the agenda at tomorrow's Cabinet meeting as ministers discuss how the Government failed to persuade people to cast their ballot.
Social Protection Minister Joan Burton today said that the Government must "look seriously" at the large number of No voters. "I think that what happened here ... there was a surge of people who decided either they didn't want to vote or that they wanted to vote No in the referendum," she said.
"In the case of the No voters I do think that the Government needs to look seriously at the reasons that people have for voting No -- that mistrust in relation to some State services," she added.
And she said that the Government should look at setting up a "more permanent form of referendum commission" to take control of the distribution of information.
Dublin North West was one of three constituencies in the country that delivered No votes, as well as the two Donegal constituencies.
The result was revealed just days after Children's Minister Frances Fitzgerald was left red-faced when the Supreme Court ruled that the Government's information campaign was biased.
Now ministers are facing yet another headache in the form of a constitutional challenge to yesterday's outcome. Journalist John Waters and former MEP Kathy Sinnott are among those who have raised the possibility of a legal challenge.