The figures showed a reading of 50.1 in January, lower than December's 51.2.

IHS Markit's Chris Williamson said the results indicated that the UK economy "is at risk of stalling or worse".

He said this was because of growing Brexit uncertainty coinciding with a wider slowdown in the global economy.

Economists had expected a reading of 51 - a figure above 50 indicates growth.

It follows last Friday's news that the PMI figures for manufacturing fell to 52.8 last month - the second weakest reading since July 2016.

The figures revealed that UK manufacturers were preparing for Brexit by stockpiling raw materials at a record pace and that there was a risk of the sector slipping into recession.

'Political uncertainty'

Speaking about the service sector figures, Mr Williamson said: "Growth ground almost to a halt in January, matching similar disappointing news in the manufacturing and construction sectors.

"The last three months have seen the economy slip into its weakest growth spell for six years and indicate that GDP likely stagnated at the start of 2019 after eking out modest growth of just 0.1% in the fourth quarter.

"The survey results indicate that companies are becoming increasingly risk averse and eager to reduce overheads in the face of weakened customer demand and rising political uncertainty.

"Such worries were in turn most commonly linked to heightened Brexit anxiety, though wider global political and economic factors were also seen to have been taking their toll on demand."

Analysis:

Andy Verity, economics correspondent

Until very recently, what's been most remarkable is not how much of an economic impact all the Brexit-related uncertainty has had, but how little. Growth has continued. The number employed keeps hitting new records. The services sector which makes up four-fifths of our economy has continued to expand.

But, crucially, those upbeat figures on jobs and growth are lagging indicators, about two months behind reality. The purchasing managers' index is more up to date, surveying the people in firms taking financial decisions.

According to them the most forward-looking indicators - such as new orders from customers - are down.

However - before we assume the politics of Brexit is stalling the economy, it's worth noting something. Back in July 2016, right after the referendum, respondents to the survey were even gloomier than they are now.

Some economists predicted capital flight; others predicted recession. Those fears, so far at least, have proved largely misplaced.

Mr Williamson's views on uncertainty over the impact of Brexit were backed up by other experts.

Samuel Tombs, chief UK economist at Pantheon Macroeconomics, said: "Services firms could not be clearer in blaming Brexit uncertainty for the stagnation of activity in January."

He added that the figures suggested that activity could fall over the coming months, but said that similar forecasts in 1998, 2001 and 2003 "failed to materialise".

And Duncan Brock, group director at the Chartered Institute of Procurement and Supply, said: "Brexit uncertainty continues to be at the heart of the malaise, as clients delayed orders and consumers were deeply reluctant to spend under the continuing cloud of hesitation, indecision and ambiguity."