In a strong attack on the Sinn Fein president, Mr Kenny said it was time Mr Adams told the truth about his past, specifically mentioning the mother of 10 who was murdered by the IRA.
"I would love to hear you speak the truth about some elements of your past," he said.
"You might some day tell the truth about the tragedy and the remorse and about the compassion that should have been shown for Jean McConville. Maybe you might do that, Deputy Adams," he added.
Mr Adams asked Mr Kenny withdraw the remark, but the Taoiseach refused. Mr Kenny also raised the Northern Bank raid in response to Mr Adams' claims about ministers being "millionaires".
"Deputy Adams has made a disgraceful comment about the ministers here today. I assure him that none of these people was funded by the Northern Bank or by the assets of the Northern Bank," he said.
The Dail was suspended after a furious row erupted between Mr Kenny and Mr Adams over the disappearance of Ms McConville.
Mr Adams was criticising Mr Kenny, Fine Gael and Labour over the harsh measures contained in the Budget.
But Mr Kenny told Mr Adams he wished he could detail the exact details of the Sinn Fein leader's past. The spat led Ceann Comhairle Sean Barrett to suspend the house for 10 minutes. Mr Adams again protested when the Dail resumed.
Mr Adams has repeatedly denied he was a member of the IRA and rejects allegations of involvement in the murder of Ms McConville.
However, earlier this year, former IRA bomber Dolours Price described how Mr Adams ordered her to ferry captives, including Ms McConville, across the border to be murdered.
Ms McConville was the most notorious of the cases of the so-called Disappeared, victims murdered by the IRA and whose bodies were then hidden south of the border.
Mr Adams has faced repeated calls to make a Dail statement on Ms McConville's disappearance. The Sinn Fein leadership has also denied any knowledge of the Northern Bank raid in December 2004.
A leaked dispatch reports that Mr Ahern was convinced the Sinn Fein pair had known of the £26.5m (€33.3m) robbery in 2004 because they were members of the "IRA military command".
The cable is dated February 4, 2005 and is among a batch released by WikiLeaks.