Watch Obama announce a fix to the latest ObamaCare problem

Bowing to mounting political pressure, President Obama on Thursday announced that the administration would allow health insurance companies to continue offering plans that would otherwise be canceled under ObamaCare's tighter rules.

In the latest hiccup for ObamaCare, millions of Americans have or are expected to to be dropped from their existing policies, even though Obama repeatedly promised that people who liked their insurance would be able to keep it.

"I completely get how upsetting this can be for a lot of Americans, particularly after assurances that they heard from me that if they had a plan that they like, they could keep it," Obama said. "And to those Americans, I hear you loud and clear."

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"I said that I would do everything we can to fix this problem and today I'm offering an idea that would help do it," he added.

The change would allow insurance companies to re-up existing plans one year, so long as they explain why those plans don't meet ObamaCare's new standards. However, insurers would not be required to extend those plans, putting the onus on them to keep people covered.

"The Affordable Care Act is not the reason insurance companies are going to have to cancel your plan," Obama said.

Even with the announcement, there are a few proposals in Congress that would tweak the law in slightly different ways and that could still come up for votes. Though Obama said he was open to working with lawmakers on a mutual solution, he added that he would "not accept proposals that are a brazen attempt to undermine or repeal the law."

A House plan written by Rep. Fred Upton (Mich.) — the "Keep Your Health Plan Act" — would likely do just that by allowing insurers to keep offering old policies to new customers, thus draining the exchange marketplace of the new customers it needs to survive.