Richard Williamson has returned to the UK after being told to leave Argentina
A British bishop whose readmission into the Catholic Church sparked uproar has apologised for offence he caused by denying the extent of the Holocaust.
Richard Williamson said if he had known the full harm his comments would cause, he would not have made them.
He said that his opinions had been formed "20 years ago on the basis of evidence then available".
However, Jewish leaders said the bishop failed to address the issue of whether he believed the Holocaust was a lie.
Renzo Gattegna, the president of Italy's Jewish Communities, described the apology as "absolutely ambiguous".
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.
Jewish leaders were unimpressed.
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 one thing he doesn't say, and the main thing, is that the Holocaust occurred, that it is not a fabrication, that it is not a lie," he said.
Iris Rosenberg, spokeswoman of the Yad Vashem Holocaust museum in Jerusalem, said: "If he is looking to repent, he needs to admit that he was wrong in denying the truth."
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.