The signs are not propitious. Glafkos Clerides, the Greek Cypriot President, says he will attend the UN-sponsored talks in New York which begin next month only to divert charges of bad faith. Besides, going soft on the Turks could cost Mr Clerides points in next year's presidential election.
Rauf Denktas, the Turkish Cypriot leader, is uncompromising, too. He reminds his sponsors in Ankara of $600m-worth of anti-aircraft missiles which the Greek Cypriots are preparing to receive from Russia. Turkish mainlanders, who have 36,000 troops on the island, say the missiles will threaten Turkish airspace and mutter about pre-emptive action.
Into this melee has come Mr Holbrooke, who earlier this month was made Mr Clinton's special representative on Cyprus. The appointment is significant not only because Mr Holbrooke has the President's ear, but because his approach to international disputes is "results-oriented", diplomacy-speak for getting things done.
Helping to clear up Cyprus - which means setting up the bicommunal federation which both sides say they want - would advance Mr Holbrooke's claims to the position he covets most: that of Secretary of State.
Since Turkey landed troops on the island in 1974, after a coup there threatened enosis (unity) with Greece, Cypriot Turks and Greeks have lived in generally peaceful isolation from one another. This situation outrages Greek nationalists, but suits Cyprus's Turkish minority, which constitutes 18 per cent of the population but controls 37 per cent of the land.
The immediate problem is a massive build-up of arms on both sides. The Greeks' new missile system is one part of a defence upgrade which is expected to cost them - and Greece proper - $2.6bn. The Turks are busy arming too.
Persuading Turkish Cypriots to negotiate in good faith is hard, as no one save Turkey recognises Mr Denktas's Turkish Republic of Northern Cyprus.
Last week, Mr Denktas cancelled a meeting with Sir David Hannay after learning Britain's representative on Cyprus had met members of the Greek Cypriot parliament, but would meet none from its Turkish Cypriot counterpart.
The European Commission is due to begin talks on Cyprus' accession to the EU next spring. Mr Clerides knows that not going to New York would prejudice Cyprus's chances of entry. In the same way, mainland Turks know that solving the dispute is the best way to advance their own claims to membership.
What annoys Turks is the line - repeated by Sir David - that Cyprus will join the EU, whether there has been a settlement or not.