To reach the deal, France agreed to a formula where only banks holding €30 billion ($39 billion) in assets, or holding assets greater than 20 percent of their country’s gross domestic product, would be directly regulated by the central bank. Previously France had insisted that all 6,000 banks in the euro area should be closely regulated by the central bank.

Germany had sought a reduced remit that would make the job of the supervisor more manageable and faced pressure from a powerful domestic banking lobby trying to shield many small German savings banks from closer scrutiny. But Germany agreed to allow the central bank to step in and take over the supervision of any bank in the euro area at its discretion.

The Germans also had concerns the central bank could be tempted to alter decisions on monetary policy to make its supervisory job easier, and they wanted to give the system some accountability. As a compromise, Germany agreed to give member states greater scope than originally foreseen to challenge central bank decisions.

Britain, which remains outside the 17 European countries that form the euro zone, had been seeking assurances that it could be exempt from orders from the new supervisor that would affect its banks operating abroad and lenders operating in the City of London. Britain agreed to a formula that should allow it and other countries outside the system to block most, but probably not all, decisions on rule making taken by the E.C.B., and to oppose decisions in cross-border banking disputes it disagrees with.

Britain’s chancellor of the Exchequer, George Osborne, told reporter he had ensured “that the countries that weren’t going to join the banking union, like Britain, were protected and their interests were protected.”

For countries like Spain and Ireland, the supervisor is a prerequisite for allowing them to tap a European bailout fund and inject rescue aid directly into their troubled banks. That would allow those governments to avoid weighing down their national balance sheets with yet more debt.

But the system for the direct recapitalization of banks is only likely to go ahead only once the supervisor is fully operating, and well after a German general election in October. German citizens have grown weary of paying most of the bill for bailouts.

Leaders also must clarify whether the system should apply in cases, like Spain and Ireland, where banks ran into problems before the introduction of the single supervisor.