Updated 7:33 PM ET
GENEVA High-stakes adjourned on an upbeat note Wednesday, with the European Union's top diplomat calling them "very important." Iran's foreign minister spoke of a possible "new phase" that would ease a decade of tensions over fears that his country wanted a nuclear bomb.
EU foreign policy chief Catherine Ashton did not go into details on the substance of the talks. But she read a statement endorsed by both Iran and the six countries, calling the talks "substantive and forward looking."
Confirming Iranian media reports, the statement said the two sides will meet again in Geneva on Nov. 7-8.
Ashton, who convened the talks, spoke of a "very intensive and I think a very important meeting."
Iranian Foreign Minister Mohammad Javad Zarif, who headed the Iranian side, said he hoped the results achieved over two days of talks ending Wednesday "will hopefully be the beginning of a new phase" in relations between Iran and the six powers - the United States, Russia, China, Britain, France and Germany.
"We have reached a serious stage in the talks," he told reporters.
The White House said the proposal addressing Iran's nuclear program is useful and contains a, quote, "level of seriousness and substance that we have not seen before."
White House spokesman Jay Carney said not to expect a prompt breakthrough in the high-level nuclear talks between Iran and six world powers.
Reporting from Geneva, CBS News correspondent Elizabeth Palmer said the talks got very serious very fast. In fact, an administration official told CBS News they had never had such candid, intense and straightforward talks with the Iranians. A lot of that has to do with Zarif. He was so ill with a bad back that he had to fly to Geneva flat out on his plane. Once he got to the city, he got busy and conducted the talks in English and it helped things along.
Palmer further reported that none of the details have leaked. The negotiators say it's top secret and has to stay that way because these talks have so much riding on them.They broke up saying there is lots to keep talking about. They're coming back at it on November 7.
By the measure of talks in the past, Palmer earlier reported they were moving at "." She added as further evidence that there's a substantial discussion taking place that the U.S. lead negotiator, Wendy Sherman, met privately for an hour on Tuesday with the deputy Iranian negotiator.
The lack of immediate details on what was achieved, however, made it difficult to evaluate the amount of progress made in what has been a decade of deadlocked negotiations over Iran's nuclear program.
Iran says it's not interested in having nuclear weapons. Its proposal Tuesday to the six powers focused on their demands that Iran's uranium enrichment program and other activities that could be used to make nuclear arms be stopped or reduced.
Iran wants painful international sanctions to be lifted in exchange for possible concessions it had been previously unwilling to consider. Those could be increased international monitoring of its nuclear program and the scaling back of its uranium enrichment plans - a potential pathway to nuclear arms and the centerpiece of its impasse with the West.
Tehran insists it has no interest in weapons production. Still, it has resisted both enticements and sanctions from world powers designed to force it into ending uranium enrichment and other activities that could be used to make weapons.
But negotiations now appear to be driven by the new wind generated since reformist President Hassan Rouhani was elected in June.
A member of one of the delegations at the talks said the new Iranian plan offered reductions in both the levels of uranium enrichment being conducted by Iran and the number of centrifuges doing the enrichment - a key demand of the six powers.
An Iranian official said any plan would be implemented in three stages, lasting from six months to a year.
Both men demanded anonymity because they were not allowed to discuss the confidential plan.
Iranian state TV, which closely reflects government views, said Tehran offered to discuss uranium enrichment levels. The report also said Iran proposed adopting the additional protocols of the U.N.'s nuclear treaty - effectively opening its nuclear facilities to wider inspection and monitoring - if the West recognizes Iran's right to enrich uranium.
But the Iranian official said any acceptance of the protocols would be one of the last steps in implementing the plan.