* FAO's food price index falls 1 pct to 211.3 points in June
* FAO/AMIS raise estimates for wheat, maize output in 2013/14
* Prices to be lower, more stable in new season-FAO economist (Adds detail)
ROME, July 4 (Reuters) - Global food prices fell 1 percent in June due to improving supply prospects, the United Nations' food agency said on Thursday, raising forecasts for wheat and maize output in the new season.
Food prices spiked during the summer of 2012 due to a historic drought in the United States but prospects for a rebound in global grain supply and good weather forecasts are now weighing on markets.
The Food and Agriculture Organisation's (FAO) price index that measures price changes for a basket of cereals, oilseeds, dairy, meat and sugar, fell for a second month running to 211.3 points in June, its lowest level since February.
"Crop prospects are even better than what we anticipated last month, while demand is subdued," said FAO senior economist Abdolreza Abbassian.
"We should be in for a season with lower prices and as inventories are being rebuilt, especially for corn, prices should be more stable," he added.
FAO and the agriculture body AMIS raised their estimate for 2013/14 world wheat output by 2 million tonnes to 704 million tonnes, pointing to improving prospects in nearly all major producing countries with the exception of the United States.
The organisations also raised their estimates for maize output in 2013/14 to 972 million tonnes from a previous forecast of 963 million tonnes in June.
The fall in FAO's index was driven above all by a 4 percent drop in dairy costs from high levels reached in April and May, and a 3 percent decline in sugar prices on the back of improved supply prospects.
Its cereal prices sub-index fell about 1 percent while prices of edible oils reached a six-month low.
FAO's headline index hit a peak of 238 points in February 2011, when high food prices helped drive the Arab Spring uprisings in the Middle East and North Africa.