* ECB to take on supervision of banks from mid-2014
* Lawmakers approve EU scheme to tighten policing of finance
* EU Parliament's Schulz - Cyprus shows ECB supervision alone insufficient (Writes through, adds comments from Schulz, detail)
By John O'Donnell and Claire Davenport
BRUSSELS, March 19 (Reuters) - The European Union agreed on Tuesday to let the ECB police euro zone banks, taking its first step towards a banking union just as a levy imposed on Cypriot savers showed that the bloc will struggle to respond in a united way to bank problems.
The European Parliament and member states's representatives sealed an accord reached late last year to give the European Central Bank (ECB) powers to supervise euro zone banks from the middle of next year.
That agreement had initially been applauded as a step towards integration. But the surprise levy on Cypriot bank deposits agreed as part of the country's bailout deal at the weekend has dented confidence that Europe will be united in tackling bank problems rather than leaving countries to struggle alone.
"The deeply distressing problems faced by Cyprus show how insufficient this step is in itself," said Martin Schulz, the German president of the European Parliament, calling for an EU-wide scheme to close failing banks and guarantee deposits.
Under the deal, banks that have assets of 30 billion euros or more than one-fifth of their country's economic output will be overseen by the ECB rather than national supervisors.
The ECB will also be allowed to intervene if it sees problems in smaller banks as well, giving it the clout to trigger the closure of struggling banks.
Although designed chiefly for the 17 countries in the euro zone, non-euro countries that sign up for ECB monitoring will be given equal rights of representation in the system.
Countries that wish to stay out such as Britain will also be given protection from interference from the ECB through a special voting scheme to be used when supervisors across the entire 27-member European Union meet.
The agreement, which will later be rubber-stamped by all EU countries, will also allow the European Banking Authority to conduct and publish annual stress tests on banks, one EU official said.
"This is a fundamental step towards a real banking union which must restore confidence in the euro zone's banks," said Michel Barnier, the European commissioner in charge of regulation.
The next pillar of the banking union will be the creation of a central system to close troubled banks and a fund to cover the costs, ensuring that individual countries such as Cyprus or Ireland are not left to shoulder these burdens alone.
But the reluctance of Germany and other economically strong countries to underpin such a fund will make it hard to set up.
Paul De Grauwe, an economist with the London School of Economics, said that the levy to be imposed on Cypriot savers illustrated the lack of support for pooling national resources.
"This is almost a fatal blow to banking union," he said. "The one key element of banking union is a system to help each other out and share the cost when there is a banking crisis in one country. There is no willingness to do that."
Nicolas Veron of the Peterson Institute for International Economics think tank in Washington said the Cypriot "default on insured deposits" undermined efforts to establish a common euro zone approach to dealing with failing banks.
"Clearly we don't have a banking union yet, even if we will have single supervision," he said. "Deposit guarantee is a central plank of that. I hope that lessons will be learned from this episode."
Without a fund or backstop to tackle problem banks wherever they arise, there is little chance of breaking the "doom loop" that can drag banks and governments down together.
A separate pledge by euro zone leaders to enable the bloc's rescue mechanism to directly recapitalise struggling banks once the ECB has taken on supervision also appears to be unravelling.
Again, euro zone paymaster Germany is worried it could be forced to foot the bill for struggling banks across the region.
Finland, the Netherlands and Germany want any banking problems that arise before the ECB takes over as supervisor to be disqualified when it comes to applying for assistance from the European Stability Mechanism.
The new role of bank supervisor for the ECB has worried some at the central bank, who fear it will interfere in its work as guardian of monetary policy.
Such a conflict of interest would arise if the ECB were to keep interest rates low in order to help struggling banks that it is also responsible for supervising.
Dutch ECB policymaker Klaas Knot expressed his reservations on Tuesday.
"Quite frankly, if we have arrived in the European Union in quieter waters, let's say 10 years down the road or so, I would clearly be open to re-evaluating this decision, because there continues to be a fundamental tension between monetary policy responsibilities and supervisory responsibilities." (Additional reporting by Robin Emmott in Brussels and Paul Carrel in Frankfurt; editing by Ron Askew and Rex Merrifield)