The firm's Chief Economist Chris Williamson said the strong end to 2013 in both service and manufacturing work in Britain would put economic growth at 1.9 percent, which would be the fastest pace since 2007.
The service sector for Britain "continued to expand strongly as 2013 came to an end," Markit said. But the Purchasing Managers Index that had hit 60 in November, came down to 58.8 in the survey taken Dec. 5-19.
The headline index, marking growth in the sector with figures above 50, is also down from an October reading of 62.5, which was the highest level since May 1997.
But confidence among those responding to the survey is at its highest point in nearly four years, Markit said.
The business expectations index that measures confidence in economic conditions in the next 12 months is at a level unseen since March 2010.
"Service providers ended 2013 in a buoyant mood," Williamson said in a statement.
"Although growth of business activity slowed, it's come down from super-strong levels and the pace of expansion remains elevated. Combined with still-strong growth in manufacturing and services, the PMI surveys suggest that the pace of economic growth will have accelerated in the fourth quarter from the 0.8 percent expansion seen in the third quarter," he said.
"If the buoyancy of the surveys in the fourth quarter is born out by the official data, the economy will have grown by 1.9 percent over the course of 2013; which would be the best we've seen since 2007."