Kenny said Fine Gael would name his successor no later than June 2.
Aoife Heffron, spokesperson for BoyleSports, said: "With Enda Kenny's retirement coming into effect, we have seen an influx of bets on Leo Varadkar, who was once priced at 11/10 to take over as Fine Gael leader, but now heads the betting at the short odds of 2/5".
While Noonan and Kenny have been widely praised for their management of the economy, their Fine Gael party lost a quarter of its seats at elections a year ago as voters complained they were not feeling the recovery.
Thanking those in the Fine Gael party, his constituents and family he announced he would be stepping down with immediate effect.
"I want to lead a party and a country that can heal numerous divisions that have opened up over the last decade or so".
No formal announcement was made despite rumours to the contrary that party strategists were looking at dates in the second and third week of June.
Dubliner Varadkar is now the clear favourite with the bookies to succeed Enda Kenny as Fine Gael boss, though political sources say it is way too early to call.
Kenny was due to step down at midnight to become acting party leader until a successor is chosen.
Kenny tweeted a statement as that meeting began, confirming his plans, ahead of the news being leaked in other ways.
Mr Varadkar is expected to formally launch his campaign for the Fine Gael leadership on Saturday. "One of the fundamental tenets of our democracy is a commitment to public service and the Taoiseach [PM] has more than fulfilled that obligation".
The contest will be decided using a weighted voting system across three electoral colleges - the parliamentary party, councillors and party members.
Minister Paschal Donohoe has declared his support for Leo Varadkar in the Fine Gael leadership race. "Having inherited a country and an economy that was decimated by mismanagement, Enda's strong leadership and ability to make hard decisions was recognised both at home and overseas and saw our worldwide reputation restored and strengthened", she said.
Rank-and-file members are allocated 25 percent and councilors 10 percent.