* Greece to get 10.3 bln euros new cash, 7.5 bln of it in June

* IMF willing to join bailout as euro zone offer debt relief terms

* Dijsselbloem hails "major breakthrough", new confidence at table

* Optimism that vicious circle broken - Greece's Tsakalotos

(Adds fresh quotes, market reaction)

By Jan Strupczewski, Francesco Guarascio and Alastair Macdonald

BRUSSELS, May 25 (Reuters) - The euro zone gave Greece its firmest offer yet of debt relief in what finance ministers called a breakthrough deal that won a provisional commitment from the IMF to return to taking part in the bailout for Athens, heartening investors.

After talks that lasted into the small hours of Wednesday, Eurogroup finance ministers gave a nod to releasing 10.3 billion euros ($11.5 billion) in new funds for Greece in recognition of painful fiscal reforms pushed through by Prime Minister Alexis Tsipras's leftist-led coalition, subject to some final technical tweaks.

But a bigger step forward was a deal under which the euro zone agreed to offer Athens debt relief in 2018 if that is necessary to meet agreed criteria on its payments burden. In the meantime, the currency area's rescue fund was given approval to take steps to smooth out Greece's debt service path.

However, German Finance Minister Wolfgang Schaeuble avoided any immediate commitment to rescheduling Greek debt that would have required him to secure approval from a sceptical parliament in Berlin before a general election next year.

The deal was nevertheless to secure agreement in principle from the International Monetary Fund to rejoin the euro zone in funding the bailout of Greece, subject to its board's approval.

"We achieved a major breakthrough on Greece which enables us to enter a new phase in the Greek financial assistance programme," Eurogroup President Jeroen Dijsselbloem, the Dutch finance minister, told a 2 a.m. news conference.

"It was difficult because we are asking a lot of the Greeks, the IMF was asking a lot of us, and we were asking quite a lot of the IMF to step back in," he told reporters on arrival for Wednesday's session of all 28 EU finance ministers.

Financial markets welcomed the agreement, which averted any repeat of last year's Greek default to the IMF that took it to the brink of exit from the euro area, threatening wider destabilisation of the 19-nation currency zone.

Greece's 10-year government bond yield fell to a six-month low of 7.09 percent and 2-year yields slid below 7 percent on the news. Yields on government bonds issued by Spain, Italy and Portugal - known as "peripheral" euro zone economies - also dropped as Greece's progress boosted investors' willingness to buy other riskier assets.

"The agreement between Greece and its creditors is positive for risk sentiment and in turn peripheral bond markets," said Rene Arecht, a derivatives market analyst at DZ Bank.

Acknowledging the "political capital" European ministers invested to reach the deal - a nod to strong German objections to debt relief - Dijsselbloem called it a "new phase" in a six-year drama to stabilise Greece's finances that had taken the euro zone to the brink of break-up.

Mutual trust was returning to the talks, he said, nearly a year after Tsipras's rejection of austerity measures pushed Athens close to be pushed out of the euro.



GREEK OPTIMISM

Greece will get most of the next installment of bailout funds in July to redeem bonds held by the European Central Bank and repay IMF loans, as well as starting to clear arrears in government payments to the private sector, with the rest paid after the summer.

Athens has long complained that austerity and reform measures demanded by its international creditors since its first bailout in 2010 have only deepened its long recession.

But Tsipras's finance minister, Euclid Tsakalotos, believed the cycle could now be broken. "I think there is some ground for optimism that this can be the beginning of turning Greece's vicious circle of recession-measures-recession into one where investors have a clear runway to invest in Greece," he told reporters as he left the Brussels meeting.

The IMF has long insisted on the European governments taking a hit now on the debt Greece owes them to relieve Athens of some of its burden and make its public finances more sustainable. The refusal of Germany and others to do that had led to months of wrangling with the IMF in which Athens had been something of a spectator in negotiations.

While the Europeans did not offer immediate debt relief, or make an unconditional promise of reducing the payments Athens must make to them, they did spell out criteria for stretching out maturities on Greece's loans and the grace period before it has to start paying interest on them.

Greek gross financing needs show be kept below 15 percent of its annual economic output in the medium term and below 20 percent beyond that.



IMF ON BOARD

IMF European director Poul Thomsen said he believed the measures would "deliver the necessary debt relief", though he cautioned that it was still up to the IMF board in Washington to determine whether to agree with his assessment. The extent of debt relief that would take place was still not clear, he said.

"It will deliver debt sustainability according to our standard criteria," Thomsen said, insisting that the IMF had not eased its insistence that it would lend no more to Athens unless its European creditors ease its debt burden. "I do not see this as a weakening of the debt relief proposals," he said.

But he acknowledged that the Fund made a big concession by agreeing that the debt relief would be decided only in 2018, rather than up-front, as the IMF initially demanded.

The easing of Greece's debts could be achieved by various methods, including extending some repayment maturities, the euro zone agreed - not through a "haircut" reducing the amount of nominal debt.

Germany has been insistent that the IMF should take part in the bailout because of the Fund's reputation for fiscal rigour. However, it has also resisted demands from Washington for debt relief, a move that Berlin fears would create a "moral hazard", giving euro zone debtors an incentive to break with austerity reforms.

Slovak Finance Minister Peter Kazimir, who has long been sceptical of help for Greece, said: "It was a complicated birth tonight. It's probably about as good as it gets."

Socialist French Finance Minister Michel Sapin heaped praise on Tsipras for pushing painful reforms through parliament in order to unlock the first installment of new money worth 7.5 billion euros next month, with another 2.8 billion to come.

"Even if the discussions were long, the atmosphere was always extremely relaxed," he said. "This deal is first and foremost a declaration of confidence in today's Greece." ($1 = 0.8972 euros) (Additional reporting by Philip Blenkinsop in Brussels and Dhara Ranasinghe in London; Editing by Paul Taylor and David Stamp) ((alastair.macdonald@thomsonreuters.com; Reuters Messaging: alastair.macdonald.thomsonreuters.com@reuters.net))