The easing from November in the two Purchasing Managers' Indices (PMIs) out on Tuesday confirmed analysts in their belief that the European Central Bank and Bank of England will keep interest rates unchanged at meetings later in the week.
Analysts expect the index to improve to 61.3 in December, well above the 50.0 which separates growth from contraction, from 60.1 in November.
The Reuters Eurozone Services survey of more than 2,000 companies dashed expectations for a sixth month of accelerating growth in the sector, which accounts for about two thirds of the economy in the 12-nation bloc, and showed companies remain reluctant to take on extra staff.
The headline business activity index slipped to 56.6 in December from November's 37-month high of 57.5, with falls in all three of the biggest economies - Germany, France and Italy. The consensus forecast was 57.8.
"It looks a little bit disappointing at the headline level," said James Knightley at ING Financial Markets in London.
However, NTC said the overall pace of expansion suggested the eurozone private sector had experienced its strongest growth for three years in the fourth quarter of 2003.
In Germany, there was a further sign that the economy is over the worst when the DIW research institute forecast an acceleration in economic growth this year to 1.4 percent after virtual stagnation last year.
However, analysts remain convinced that the ECB will keep its key interest rate at 2.0 percent at its first rate-setting meeting of 2004 on Thursday.
In Britain, the main activity index of the Chartered Institute of Purchasing and Supply/Reuters monthly report on services, which covered around 700 companies, fell to 58.5 from a 6-1/2 year high of 59.6 in November.
The headline number came in slightly below the 60.0 that the market had expected.
"It is a bit disappointing relative to market expectations but it is important not to blow it out of all proportion - it is still a very strong number," said Richard Batley, economist at Halifax, Britain's biggest mortgage lender.
The fall in British activity was caused partly by slower growth in new orders and outstanding businesses.
The slowdown is unlikely to change the Bank of England's perceptions about the British economy when it meets on Wednesday and Thursday.