Opinion polls say voters will likely back the plans to scrap the Seanad, or Senate, which Kenny believes is elitist and ineffective, but turnout figures yesterday appeared low before polls closed at 0230 IST.
The count begins today and the results are expected over the weekend.
In the Dublin North West constituency, just over 22 per cent of the electorate had cast their vote by 0100 IST, while in Dublin Central 24 per cent cast their ballots and in Dublin North Central 39 per cent.
Political commentators expect a similar national turnout to the last referendum when just 33.5 per cent of the electorate cast their vote in November last year.
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The cost-saving proposals comes after Dublin was forced to accept an EU/IMF bailout in late 2010, with many blaming the country's politicians for failing to properly manage the "Celtic Tiger" economic boom.
"Other small countries like Sweden and Denmark have clearly shown that single chamber parliaments not only cost less but they work much more effectively and with far greater transparency," he said.
"After 70 years of no change, it is time to save money, put the public ahead of politicians and abolish the Seanad."
Opponents of Kenny's plans admit the 60-member upper house in its current form does not work but have said it should be reformed rather than closed.
The move was criticised as pandering to voters angered by the high cost of government in a country still recovering from a painful economic collapse.
Fine Gael claim abolishing the upper house will save 20 million euros (USD 27 million).
A separate vote on whether to establish a new Court of Appeal also took place, with Dublin hoping the new institution will ease the heavy pressure on the heavily backlogged Supreme Court.