* Prospects for light storage build next week lifts prices * Mild forecasts for late this week, next week limit gains * Tropical activity stirs in the Atlantic basin By Joe Silha NEW YORK, Sept 13 (Reuters) - U.S. natural gas futures ended higher on Friday, backed by this week's supportive inventory report and expectations for a light build next week, but the upside was limited by the milder weather expected next week which should slow demand. Some traders viewed Thursday's 65 billion cubic feet weekly inventory build as supportive for prices, noting it came in slightly below the Reuters poll estimate of 66 bcf after exceeding consensus expectations the previous two weeks. "People are expecting a supportive weekly inventory build next Thursday after all the heat this week, and the tropics may have generated a little (short) covering today ahead of the weekend," said Steve Mosley at The SMC Report in Arkansas. Front-month gas futures on the New York Mercantile Exchange ended up 3.9 cents, or 1.1 percent, at $3.677 per million British thermal units, after trading between $3.603 and $3.689. Front futures hit a six-week high of $3.719 last week. The October-January futures spread, which on Thursday settled near a two-month low of 30.5 cents (January premium), ended Friday at 31.2 cents. A narrow spread reflects views that inventories will be very comfortable by winter. Storage gas is a key source of supply to help meet peak heating season demand. The nearby contract, which fell 1.4 percent last week in its first weekly decline in four weeks, finished this week up 4.2 percent, backed by strong heat that kicked up demand. Technical traders said the front month close above resistance at the 200-day moving average in the $3.67 area could send prices higher next week, but most needed another higher close to predict more upside. Recent gains have not come easily as traders also focus on the milder weather ahead that should slow demand. Many remain skeptical of the upside in the near term, with summer heat tapering off, inventories on the rise and production flowing at or near a record peak. Forecaster AccuWeather.com expects temperatures in the Northeast and Midwest, key gas consuming regions, to mostly range from normal to below normal for the next 10 days. U.S. Energy Information Administration data on Thursday showed that total domestic gas inventories stood at 3.253 trillion cubic feet last week. Stockpiles are about 5 percent below last year's record highs at that time, but 1.4 percent above the five-year average. Early injection estimates for next week's report range from 55 bcf to 68 bcf. Stocks gained 61 bcf during the same year-ago week. The five-year average increase for that week is 74 bcf. Baker Hughes data on Friday showed the gas-directed rig count rose by seven this week to near a six-month high of 401. It was the second weekly gain for the gas rig count, which posted an 18-year low of 349 in late June. Recent gas rig count gains - the count has risen in eight of the last 12 weeks - have stirred views that new investment in gas pipelines and processing plants is allowing producers to pump even more supply into an already well-supplied market. The EIA still expects U.S. gas production in 2013 to hit a record high for a third straight year. In the ICE cash market, gas for weekend delivery at Henry Hub , the benchmark supply point in Louisiana, rose 3 cents to $3.60, but late Hub differentials weakened to about 3 cents over NYMEX from a 6-cent premium on Thursday. Gas on the Transco pipeline at the New York citygate was unchanged at $3.72 despite the mild weekend outlook, while Chicago was 2 cents higher at $3.69. The U.S. National Hurricane Center was monitoring four tropical systems in the Atlantic basin, but traders noted there were no serious threats to Gulf of Mexico gas production.