* Progress at climate U.N. talks
* Promised greenhouse gas cuts by rich fall short
* Rich countries say private sector to provide funds
(Updates with closing remarks by U.S., EU)
By Alister Doyle and Gerard Wynn
BONN, Germany, June 12 (Reuters) - Climate talks ending on Friday made progress towards a new U.N. treaty to curb global warming but fell short of calls by developing nations for the rich to make deep cuts in greenhouse gas emissions.
Four years of talks to widen the existing Kyoto Protocol have struggled to agree how to share the cost. The United States and Europe warned in closing remarks in Bonn that the private sector would finance the climate fight, not their governments.
"I look back on this as a significant session that has advanced our work in important ways," Yvo de Boer, head of the U.N. Climate Change Secretariat, told a news conference of the June 1-12 meeting among 183 nations.
He said governments staked out far clearer views after a first review of a draft legal text of a treaty due to be agreed in Copenhagen in December to succeed the Kyoto Protocol. It will curb greenhouse gases mainly emitted by burning fossil fuels.
Developing countries called for more.
"We finally managed to have a positive exchange on the numbers" for developed nations, China's climate ambassador Yu Qingtai said. China and the United States are top emitters.
"But still we hear repeated statements resisting calls for further meaningful cuts," he told Reuters.
China and many developing nations want the rich to cut by at least 40 percent below 1990 levels by 2020 to avoid the worst of global warming such as droughts, floods and rising sea levels.
Poor countries say developed nations got rich on industrialisation which spewed greenhouse gases into the air, and want to be paid to avoid the same high-carbon growth.
The United States and Europe poured cold water on hopes of substantial public funds, such as 1 percent of national wealth or more as demanded by many developing countries.
"The key issue is not the number," said Jonathan Pershing, head of the U.S. delegation, referring to "marginally" bigger investments to improve efficiency or to install low-carbon instead of polluting coal plants.
"We'd like to change that" view of developing countries that governments would bankroll the fight against climate change, he said, adding carbon offset markets could play a big role. The European Union also underscored private finance would dominate.
Pershing said progress in Bonn had been "slow", and the European Commission's Artur Runge-Metzger said "enormous effort" was required to get a deal in Copenhagen in December.
The United States expected China to undertake actions, such as renewable energy targets, but not be be legally bound to prove curbs on emissions. China and the United States are the top two emitters. "We have advanced perhaps a couple of miles towards Copenhagen. We still have thousands to go," said Jennifer Morgan of the London-based E3G think-tank.
Outside the talks in a Bonn hotel, protesters brought along two live camels and laid out some sand to illustrate fears of creeping desertification. "We spit on weak targets," one banner said, another said: "Shrinking targets, growing deserts".
The chair of a group looking at new actions to curb emissions by all countries said a draft text had swollen with new ideas from about 50 pages to 200. Big breakthroughs were likely to happen only in Copenhagen, he said.
"This is like the evolutionary process in reverse. The Big Bang comes at the end," said Michael Zammit Cutajar, of Malta.
-- For Reuters latest environment blogs click on: blogs.reuters.com/environment/ (Editing by Richard Balmforth)