It ends four months of uncertainty for the troubled lender's four million customers after it was formally put up for sale.
The bank said the deal would enable it "to thrive as a stand-alone entity, with values and ethics and strong customer services continuing at the heart of its business".
The announcement confirmed an exclusive report on Sky News that revealed key details of the rescue deal - its second since 2013 when the hedge funds took control of the lender following the bank's near collapse under Co-op Group ownership.
The agreement will see the consortium of Wall St financiers put up around £700m of new money to shore up the bank's balance sheet and meet the capital requirements of regulators.
It also plans to raise, partly through the sale of new shares, hundreds of millions of pounds in new money.
A major sticking point in the negotiations had been tensions over the division of the £10bn pension scheme shared by the Co-op Group and Bank, and the ongoing relationship agreement between them.
It was confirmed by the bank that it had agreed to separate itself from the scheme, leaving the lender solely responsible for its employees' retirement savings.
Under the deal, the Co-op Group's 20% stake in the bank would fall from its current 20% level to 1% in the new holding company.
Bank chairman, Dennis Holt, said: "The board is pleased to confirm this proposal for a recapitalisation which will mean that the Co-operative Bank can continue as a viable stand-alone entity, with values and ethics at its heart.
"It is a great outcome for our customers. Our investors share our commitment to building our distinctive ethical franchise and see strong future growth potential for The Co-operative Bank."
The Co-op Group said it "welcomed" the deal while a spokesman for the Prudential Regulation Authority said: "The PRA has accepted the Co-operative Bank's plan to build greater financial resilience.
"Supervisors will remain closely engaged with the bank while the actions announced today are taken forward."