BEIJING, Sept 1 (Reuters) - China's vast factory sector grew at its slackest pace in three months in August as growth in output and new orders cooled, a private survey showed, adding to worries that the economy is losing momentum despite government steps to steady it.

The final HSBC/Markit Manufacturing Purchasing Managers' Index (PMI) slipped to 50.2 in August - just a whisker below a preliminary reading of 50.3 - from July's 18-month high of 51.7.

It was the lowest reading since May, though the PMI stayed for a third consecutive month above the 50-point level that separates growth in activity from contraction.

A sub-index measuring new orders, a gauge of demand at home and abroad, fell to a three-month low of 51.3 while the sub-index for output eased from July's 16-month high.

(Reporting by Xiaoyi Shao and Koh Gui Qing; Editing by Richard Borsuk)