SHANGHAI, (Reuters) - Activity in China's services sector expanded at its slowest rate in more than a year in August, a private survey showed on Tuesday, a worrying sign for what has been one of the lone bright spots in the cooling economy.

The Caixin/Markit Purchasing Managers' Index (PMI) fell to 51.5, its lowest level since July 2014, from July's reading of 53.8.

A reading above 50 points indicates growth on a monthly basis, while one below that points to a contraction.

New business rose in August but at a milder pace, making firms more cautious about hiring. Employment in the sector fell to 50.1, its lowest level since August 2013.

Services have accounted for the biggest part of China's economic output for at least two years, with resilience in the sector helping to cushion the downdraft from persistent weakness in the factory sector.

The government has taken a series of steps since last year to try to shore up growth, including accelerating infrastructure spending and repeated reductions in interest rates and banks' reserve ratios.

But the official growth target for this year is still expected to moderate to around 7 percent, the slowest in a quarter of a century.

After a series of disappointing economic data and twin currency and stock market shocks, some economists say current growth levels could already be much weaker, and further easing measures are widely expected.

(Reporting by Sue-Lin Wong; Editing by Kim Coghill)