* U.S. export sales up 36 pct from the previous week

* ICE inventories fall to lowest level since October

Dec 19 (Reuters) - Cotton futures edged up on Thursday in light volumes, underpinned by strong weekly U.S. government export data and a sense of tight U.S. supplies even as a recent price rally kept mill buyers largely on the sidelines.

The most-active March cotton contract on ICE Futures U.S. closed up 0.33 cent, or 0.4 percent, at 83.33 cents a lb.

Weekly U.S. government data showed exporters sold 236,000 running bales in the week ending Dec. 12, up 36 percent from the previous period.

Turkey and China buyers remained active even as the benchmark contract broke above 80 cents per lb for the first time in a month.

Worries have grown over tight U.S. supplies in the 2013/14 season that began on Aug. 1, throwing the market into a backwardation, where spot prices are trading above second-month prices, in the last seven sessions.

The United States is wrapping up the harvest and ginning of its smallest crop in four years.

Further, exchange inventories have fallen to 41,714 bales, at the lowest level since October and down steeply from above 225,600 bales at the start of the month, the most recent ICE data compiled by Reuters showed.

"If we keep selling at this pace, we won't have any cotton. The price has to stay here or go higher to ration demand," said Ron Lawson, managing director of logicadvisors.com.

Even so, mill buyers remained on the sidelines, buying in a hand-to-mouth fashion and capping gains.

(Reporting by Chris Prentice; editing by Andrew Hay)