The bishop said he did not mean to cause offence
The Vatican has rejected an apology by a British bishop who denied the full extent of the Holocaust.
It said the bishop needed to "unequivocally and publicly" withdraw his comments.
Earlier, Jewish leaders said the bishop had failed to address the issue of whether he believed that the Holocaust was a lie.
Richard Williamson said if he had known the full harm his comments would cause, he would not have made them.
The bishop said that his opinions had been formed "20 years ago on the basis of evidence then available".
But Vatican spokesman Rev Federico Lombardi said the bishop "does not seem to respect the conditions" it set after he had made the comments.
Meanwhile Renzo Gattegna, the president of Italy's Jewish Communities, described the apology as "absolutely ambiguous".
Rabbi Marvin Hier, founder and dean of the Simon Wiesenthal Center in Los Angeles, said the statement "is not the kind of an apology that would end this matter" because it failed to address the central issue.
The row began after Bishop Williamson was reinstated into the Church.
The Vatican has been embroiled in an international row after Pope Benedict XVI lifted an excommunication order on the British bishop in January.
Church leaders said the Pope had not been aware at the time of an interview given by the bishop, a member of the traditionalist Society of St Pius X (SSPX), to a Swedish TV programme last November.
In it, he disputed that six million Jews had been killed by the Nazis, and said that none had died in gas chambers.
The Pope has since called on Bishop Williamson to recant his views.
In a statement published on the SSPX website, the bishop says his superior, Bishop Bernard Fellay, and the Pope "requested that I reconsider the remarks I made on Swedish television four months ago, because their consequences have been so heavy".
"Observing these consequences I can truthfully say that I regret having made such remarks, and that if I had known beforehand the full harm and hurt to which they would give rise, especially to the Church, but also to survivors and relatives of victims of injustice under the Third Reich, I would not have made them."
He added: "On Swedish television I gave only the opinion
of a non-historian, an opinion formed 20 years ago on the basis of evidence then available and rarely expressed in public since.
"However, the events of recent weeks and the advice of senior members of the Society of St Pius X have persuaded me of my responsibility for much distress caused. To all souls that took honest scandal from what I said before God I apologise."
The BBC's religious affairs correspondent Robert Pigott says the apology stops short of fully recanting the bishop's earlier statements.
The bishop was one of four ultraconservative SSPX bishops whose excommunications were lifted by Pope Benedict XVI on an unrelated matter.
He is back in the UK after being expelled from Argentina earlier this week for concealing "the true motive for his stay in the country".
Bishop Williamson had been the director of a seminary in La Reja, but had said he was an employee of a non-governmental group rather than declaring "his true activity", Argentina's interior ministry said.