Mr Adams was criticising Mr Kenny, Fine Gael and the Labour Party over the harsh measures contained in the Budget, when Mr Kenny told Mr Adams he wished he could detail the exact details of the Sinn Fein leader's past.
Mr Kenny said it was time Mr Adams told the truth about his past, specifically mentioning the mother of ten who was murdered by the IRA.
"I would love to hear you speak the truth about some elements of your past," Mr Kenny told Mr Adams.
"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."
A furious Mr Adams asked that Mr Kenny withdraw the remark, but the Taoiseach refused. It led Ceann Comhairle Sean Barrett to suspend the house for ten minutes. Mr Adams again protested when the Dail resumed.
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.
Mr Adams has repeatedly and forcefully rejected any suggestion that he played a part in the death of Ms McConville