The Harley Medical Group, which fitted PIP breast implants to almost 14,000 British women, has said it will not replace them free of charge.
The private firm fitted more of the implants than any other UK cosmetic surgery firm.
It follows days of delay from the company in giving clear advice to women affected by the scandal.
The group claims replacing the banned implants would put it out of business.
Another private company, Transform, has also said it will not replace the implants free of charge.
Earlier this week, it said it was reviewing its options but, as it stands, women will have to pay £2,800 to have the implants removed.
Around 40,000 British women received PIP implants manufactured by the now closed French company Poly Implant Prostheses (PIP).
The implants were filled with non-medical grade silicone intended for use in mattresses.
The Government has said anxious patients who had their surgery on the NHS will be able to have the implants removed and replaced free of charge.
The NHS will also pay to remove, but not replace, implants if a private clinic refuses or no longer exists.
Today, Health Secretary Andrew Lansley reiterated the Government's view that private clinics have a moral duty to look after their patients.
He also announced further investigations into the scandal, including looking at the EU approval of the devices which resulted in them receiving a CE mark.
The actions of the Medicines and Healthcare products Regulatory Agency (MHRA) in the UK will also be analysed further.
The Harley Medical Group's chairman Mel Braham told the BBC the firm had neither the resources, the surgeons, nor the operating facilities needed to do the surgery.
He said the Government had a moral responsibility to replace the PIP implants, because of what he called the failure of regulation by the MHRA.
Other leading providers including BMI Healthcare, Nuffield Health and Spire have agreed to offer free removal of the PIP implants.
The Hospital Group has said it will only pay to replace PIP implants that have ruptured.
The group's website tells patients "to ask your GP to refer you for an MRI scan".
It said: "We are happy to have a look at the report and assess that for you.
"We are replacing all ruptured PIP implants free of charge."
Ramsay Health Care, which has around 150 patients with PIP implants, said it was offering concerned women the chance to be examined and, if there is a rupture or clinical need, the implants would be removed and replaced free of charge.
Today, Mr Lansley reiterated the views of an expert panel convened to look at the issue, saying there is no "causal link" between the implants and breast cancer.
But the panel found it was undeniably the case that the implants are made up of non-medical-grade silicone and should not have been implanted in women in the first place.
The expert group was unable to establish if the rupture rate is higher for PIP implants than for others.
Mr Lansley said the events "highlight the need to ensure the safety of people having cosmetic interventions".
He said "lessons need to be learned from this case and incorporated into the ongoing review of the EU Medical Devices Directive".
"We also need understand what happened in this instance within the UK.
"A review for this purpose will be led by the Minister for Quality... It will investigate and report rapidly.
"The blame for what happened lies with PIP, but this review will enable us to learn lessons to improve future regulatory effectiveness."
Mr Lansley repeated the pledge for the Care Quality Commission to conduct a swift review of private clinics.
This will look at evidence of compliance, patient safety and clinical quality, and the information and support given to patients.
A review has already been set up into the regulation and governance of the entire cosmetic surgery sector, led by Professor Sir Bruce Keogh, and will include treatments such as cosmetic surgery and dermal fillers.
"The review will consider issues of governance, data quality, record keeping and surveillance, as well as ensuring that proper information is provided to secure patients' informed consent," Mr Lansley said.
"I expect this to include consideration of an outcomes-based register of frequently implanted devices - covering everything from breast implants to heart valves to replacement joints - to provide the UK with a valuable asset for further innovation and safety improvement.
"There is already considerable clinical support for such a comprehensive registry."
The Harley Medical Group has 13,900 patients with PIP implants.
Mr Lansley said PIP received a CE Mark for its breast implants in 1997, overseen by the German notified body TUV Rheinland.
The MHRA's adverse incident centre received a number of reports annually about PIP implants, he added.
As a result, in 2008, the MHRA raised its concerns with the German regulatory authorities and the manufacturer.
"The MHRA were reassured by the notified body that the number of adverse reports were adequately explained by changes in the company's reporting practices and by the increasing number of implants sold," Mr Lansley told MPs in the Commons.
"However, in March 2010, the French regulator discovered that, rather than using medical grade silicone gel in the implants, PIP had in fact been using unauthorised silicone gel.
"This is in clear violation of the CE Mark.
"The CE Mark was promptly withdrawn and all EU member states immediately alerted.
"The MHRA immediately issued advice to stop using PIP breast implants in the UK."
Yesterday, the Welsh Government said it would pay to both remove and replace breast implants from women who had them fitted by a private clinic.
In France, the government has told 30,000 women they should have the implants removed, while the Czech and German authorities have recommended that women should also have them taken out.