By Padraic Halpin

DUBLIN (Reuters) - Business expectations in Ireland's services sector tumbled in June, a survey showed on Tuesday, as some firms expressed concern that Britain's vote to leave the European Union would lead to a slowdown in activity.

Sentiment slipped across sectors from tourism to telecommunications.

Ireland, Europe's fastest-growing economy, is considered to have more to lose than any fellow European Union member by its nearest neighbour and biggest trade partner's decision to quit the bloc both countries joined together 43 years ago.

The Investec Purchasing Managers' Index (PMI) of activity in services fell only slightly to 61.2 in June, from 61.7 in May, although most of the responses were given before the market turmoil that greeted Britain's referendum result.

However the sub-index measuring business expectations fell sharply to 68.4 from 77.0, the lowest mark since August 2013 when Ireland was coming towards the end of a three-year international bailout programme.

"Some panelists expressed concerns that Brexit would have a negative impact on activity, which is a view that we share," Philip O'Sullivan, chief economist at Investec Ireland, said of the survey which covers businesses from banks to hotels.

"We are unsurprised to see the unadjusted data suggest that, although still well in positive territory, confidence has fallen to a two year low in travel and leisure, and to three year lows in business services and technology, media and telecommunications."

Almost one in five Irish services firms sell into the United Kingdom and O'Sullivan said he expected a further moderation in the rate of growth over the coming months at least.

The main index has been above the 50 mark denoting growth for almost four years and June's reading remained both close to the 10-year high of 64.0 at the start of the year and comfortably above the euro zone average of 53.1.

"The survey confirms the rapid growth of the Irish economy in the period immediately prior to last month's Brexit vote," said Dermot O'Leary, chief economist at Goodbody Stockbrokers.

"That said, given Ireland's close economic links with the UK, we can expect a further slowdown in the services sector over the coming months as the Brexit effects feed through to activity."

Ireland's economy is forecast to grow faster than any other in Europe for the third straight year in 2016 at 4.9 percent, but the government has already cut its forecast for 2017 to around 3.4 percent from 3.9 percent and warned of worse ahead if Britain strikes an unfavourable post-Brexit deal with the EU.

(Additional reporting by Conor Humphries,; Editing by Jeremy Gaunt)