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CHICAGO, June 13 (Reuters) - U.S. soybean futures hit a one-month high on Tuesday on worries about dry conditions in the Midwest hurting crop prospects and spillover strength from crude oil, traders said.

Corn futures were mostly higher on the Midwest weather jitters, but the nearby July contract fell on worries about demand for U.S. supplies.

Wheat also rose, posting the day's highs after Russian President Vladimir Putin said Moscow was considering withdrawing from the Black Sea grain export deal, raising uncertainty about global grain supplies.

As of 1:03 p.m. CDT (1803 GMT), Chicago Board of Trade July soybean futures were up 27-1/2 cents at $14.00-1/4 per bushel after rising to $14.15-1/2, the contract's highest since May 12.

CBOT July wheat was up 3-1/4 cents at $6.37 per bushel. For corn, the nearby July contract was down 4-3/4 cents at $6.12-1/2 a bushel, but back months were higher, with new-crop December corn up 2 cents at $5.51-1/4.

Soybeans and deferred corn futures climbed on worries about crop stress in the United States, underscored by larger-than-expected declines in the U.S. Department of Agriculture's weekly condition ratings for each crop.

Late on Monday, the USDA rated 61% of the U.S. corn crop and 59% of the soybean crop as good-to-excellent, each down 3 percentage points from the prior week, while analysts on average had expected a 2-point drop for both crops.

"It is dry out there," said Jack Scoville, analyst with the Price Futures Group in Chicago.

Still, front-month July corn bucked the higher trend, falling in response to weak demand for U.S. corn supplies.

"The beans should start seeing some better demand come to the United States in the next couple of months. But the corn demand should stay pretty lousy because Brazil will start to offer corn, and their prices are going to be cheaper than ours," Scoville said.

Brazilian farmers will reap record soybean and corn crops this year, the government's food supply and statistics agency Conab said, citing favourable conditions in key growing regions.

Meanwhile, U.S. crude oil prices climbed more than 3% after China's central bank lowered a short-term lending rate for the first time in 10 months. Soybeans and corn sometimes follow trends in crude oil given their roles as feedstocks for biofuels. (Additional reporting by Gus Trompiz in Paris and Matthew Chye in Singapore; Editing by Shweta Agarwal and Grant McCool)