Maria Miller is facing a battle to hang on to her Cabinet job after being ordered to repay £5,800 of mortgage expenses and apologise on the floor of the House of Commons.
The Culture Secretary was heavily criticised by the cross-party Standards Committee for failing to co-operate fully with a 15-month probe into her claims.
But she was found not to have broken rules by moving her parents into her taxpayer-funded second home.
Ms Miller said in a statement: "I have accepted the Committee's report in full and I will apologise. I am pleased that the Committee has dismissed the allegation made against me by a Labour MP."
Downing Street said she had David Cameron's "very strong and very warm support".
The Prime Minister's spokesman said the approach she was taking by accepting the report and apologising was "absolutely the right one".
Mrs Miller spoke with the Prime Minister by phone this morning, Mr Cameron's official spokesman said.
The spokesman repeatedly declined to say whether she apologised personally to Mr Cameron or offered her resignation.
He said: "The matter does not arise because she is accepting the committee's report in full, she will apologise to the House. That is absolutely the right thing to do and the Prime Minister is satisfied with that outcome.
"She explained the approach that she was going to be taking. The Prime Minister thinks that is absolutely the right one. What actually was happening is that he was offering her his full, strong, very warm support."
The spokesman rejected suggestions that Mrs Miller's ministerial position was more secure because she was a woman - amid criticisms of a lack of female MPs in the upper tiers of Mr Cameron's government.
"He would not accept that at all," he told reporters.
The investigation was launched in December 2012 amid allegations that Mrs Miller had benefited by her parents living in a London home on which she had received around £90,000 in expenses between 2005 and 2009.
The committee found that Mrs Miller should "properly" have declared the property as her main family home because she was spending most nights there, and instead claimed for accommodation in her Basingstoke constituency.
But the decision was "finely balanced" and the MP had acted "reasonably in the light of the guidance available at the time".
The report made clear there was nothing in the rules preventing Mrs Miller from moving her parents into an expenses-funded home - as long as she reduced claims to take account of the situation.
Standards Commissioner Kathryn Hudson concluded that that appeared to have been done, even though she had made no "formal arrangements by which she could demonstrate transparently that she was not claiming for their costs".
The committee said Mrs Miller's financial arrangements were so "complex" it was difficult to track the changes in her borrowing over the period and admitted many records were no longer available.
When she was elected in 2005 there was a mortgage of around £420,000 on the house, but by 2009 that had risen to just below £580,000.
For the first three years she claimed between three quarters and four fifths of the mortgage interest she was paying out, according to the committee.
However, in 2008-2009 she claimed the total mortgage interest sum for several months and at other times just a hundred pounds below.
Mrs Miller suggested that she needed to repay £4,000 as a result of the "inadvertent" error, blaming sharp falls in interest rates for the confusion.
But the committee ruled that £5,800 was the right figure.
"We are concerned that Mrs Miller did not pay as close attention to the rules of the House as she should have done," the report said.
"As we have seen, after her election she increased the facility on her mortgage on at least two occasions without consulting the House, despite the fact that in both the 2005 and 2006 Green Book the advice given to those who wished to change their mortgage was: 'Please consult us in advance. There are strict rules on the costs that can be claimed, and you may need to change the nomination of your main home'.
"While Mrs Miller has consistently told us that she never intended to claim the interest on the £50,000 mortgage increase revealed by the Commissioner's initial investigation, there is no documentation as to how she apportioned her claims, and towards the end of the period in some months she not only claimed for the entire mortgage interest charged, but appears to have claimed slightly more than that interest.
"There is no indication that she considered whether or not her variable mortgage or the increase clearly shown in the RBS documentation from a facility of £425,000 to £525,000 might have engaged the prohibition against additional mortgages.
"The documentation that is available of Mrs Miller's interactions with the House tends to show a pattern in which officials would press her for information and the information that was provided appears to have been the minimum necessary."
But the report concluded: "Even though the figures available are incomplete, we are satisfied that there is sufficient independent evidence to support Mrs Miller's assertion that up until the year 2008-09 she did not claim for the interest on any increases to her mortgage after her election."
The MPs condemned Mrs Miller for her attitude to the commissioner's inquiry, saying she had breached the House's Code of Conduct by failing to co-operate fully.
"Much of the delay and difficulty in this case has arisen from incomplete documentation and fragmentary information. Mrs Miller has to carry significant responsibility for that," the report said. "She should have attempted to provide the explanation and documentation requested by the Commissioner to the Commissioner at the outset rather than requiring us to seek the information directly."
Mr Cameron made clear that Mrs Miller would not be sacked over the breach - even though she is thought to be the only minister for decades to issue such a personal apology without resigning.
"Maria Miller is doing an excellent job as Culture Secretary and will continue to do that," the PM said.
"If we look at this report, yes, of course these issues do matter but she was cleared of the original allegation made against her.
"An over-payment was found which she is going to pay back and she'll make a full apology and I think people should leave it at that."
Speaking, during a visit to Birmingham Airport, Mr Cameron added: "She has said she is going to apologise in front of the House of Commons and make clear that apology.
"But in terms of what the report found, it actually cleared her of the original allegation against her.
"It did find an over-payment which it refers to as an administrative error and it's important Maria repays that money and that's exactly what she is going to do.
"That is what this committee - remember this committee includes independent people, non-politicians - that's what they've recommended and that's what she'll do.
"I think people should leave it at that."
In a statement to the Commons lasting under a minute, Mrs Miller stressed the allegation that the taxpayer funded her parents' living costs had been "dismissed".
"The committee did recommend that I apologise to the House for my attitude to the commissioner's inquiry and I of course unreservedly apologise.
"I fully accept the recommendations of the committee and thank them for bringing this matter to an end."