By Jonathan Cable

LONDON (Reuters) - Surprisingly strong growth in France supported stable euro zone private business activity during August but factories could face a tougher September as new order growth stumbled, surveys showed on Tuesday.

Muddying the outlook for the coming months is the United Kingdom's vote in late June to leave the European Union, although so far the economic repercussions seem to have been confined to Britain, not its main trading partner.

"August's slight rise in the euro zone Composite Purchasing Managers' Index suggests that, despite shrugging off the UK's Brexit vote, economic conditions remain fairly subdued," said Stephen Brown at Capital Economics.

France's private sector shrugged off its neighbor's vote and accelerated to levels last seen just before the militant attacks in Paris in November, as an upturn in the service sector offset continued weakness in manufacturing.

Those attacks, and the recent one in Nice in July, hit the country's service industry - the hotel and restaurant sector in particular - and resulted in lower demand for travel to Europe.

In France, the travel and tourism sector's contribution to GDP will grow 1.1 percent this year, down from a previous forecast of 2.9 percent, the World Travel and Tourism Council said on Monday.

Still, the brighter overall picture should alleviate fears the French economy continued to slow down this quarter after unexpectedly stagnating in the second quarter of the year.

German private sector growth slowed in August, but remained robust overall, its PMI showed, suggesting Europe's biggest economy is set to keep on expanding in the summer months after it grew more than expected in the second quarter.

Consumer confidence fell markedly again across the currency bloc in August, a sign of low morale after the British decision to leave the EU, official data showed on Tuesday.

The European Commission's flash estimate, which defied economists' forecasts of a rebound in confidence this month, showed euro zone consumer morale decreased to -8.5 in August from an unrevised -7.9 in July.

UNDER PRESSURE

Markit's flash composite Purchasing Managers' Index for the euro zone edged up to a seven-month high of 53.3 from July's 53.2, where any reading above 50 indicates growth. A Reuters poll of economists had predicted a slight dip to 53.1.

Markit said the PMI pointed to GDP expanding 0.3 percent this quarter, matching a Reuters poll earlier this month that showed the euro zone economic outlook stable but lacklustre, about half the speed at the start of the year.

"With underlying growth remaining muted, the ECB looks set to ease monetary policy further by year-end. After all, we do not expect inflation to increase by then either," economists at Commerzbank told clients.

Pressure remains on the European Central Bank to announce more easing as it has so far been unsuccessful in getting inflation anywhere close to its 2 percent target ceiling.

It is currently at just 0.2 percent year-on-year.

But there is little confidence amongst economists about just how much firepower the ECB has left. [ECILT/EU]

Of some concern, having only trimmed their prices in July, firms returned to deeper discounting this month. The euro zone output price index fell to 49.5 from 49.8.

Discounting helped drive a PMI covering the bloc's dominant service industry up to 53.1 from 52.9, also confounding expectations for a dip to 52.8. The manufacturing PMI was predicted to have held steady at July's 52.0 but fell to 51.8.

The factory output index, which feeds into the composite PMI, nudged up to an eight-month high of 54.0 from 53.9.

However, new order growth was at its weakest since early 2015, falling to 51.5 from 52.2, suggesting the headline manufacturing PMI may decline next month.

Service firms were also less optimistic about the year ahead. The business expectations index fell to 60.2 from 60.9, its lowest reading since late 2014.

(Editing by Jeremy Gaunt)