* Gas rig count falls eight to 420 * Oil, horizontal rig counts up slightly for week NEW YORK, March 1 (Reuters) - The number of rigs drilling for natural gas in the United States fell this week for the fourth time in five weeks, as relatively weak gas prices continued to discourage dry gas drilling. The gas-directed rig count fell this week by eight to 420, data from Houston-based oil services company Baker Hughes Inc showed on Friday. The gas count is hovering just above the 13-1/2-year low of 413 posted in early November. Producers have mostly been curbing dry-gas drilling in favor of more profitable oil and liquids-rich plays such as Eagle Ford in Texas and Marcellus in Appalachia. The oil-focused rig count, which hit a 10-month low of 1,315 five weeks ago, rose by four to 1,333, Baker Hughes data showed. The oil count is still up 40 rigs, or about 3 percent, from the same week last year. Baker Hughes also reported that horizontal rigs, the type often used to extract oil or gas from shale, rose by one this week to 1,141. The horizontal count, which has risen in four of the last five weeks, is still down about 4 percent from the record high of 1,193 set in May. Drilling for natural gas has mostly been in decline for more than a year. The count is down about 55 percent since peaking in 2011 at 936, but so far production has not shown any significant signs of slowing. The associated gas produced from more-profitable shale oil and shale gas liquids wells has kept dry gas flowing at or near a record pace. Data from the U.S. Energy Information Administration on Thursday showed that gross natural gas output in December fell 1.1 percent from November's record high, the first time in four months that production failed to notch a new peak. But most analysts pegged the decline to cold weather in the Southwest that likely froze wells and not because producers were finally curbing dry gas flows. The EIA expects marketed gas production in 2013 to hit a record high for the third straight year. Gas futures prices, which showed little reaction to the report, are currently trading near the $3.45 per mmBtu area, which should be below the cost of most dry gas production.