Sales volumes on the high street also strengthened last month, according to a separate CBI survey of retailers. The biggest jumps were in footwear and clothing, possibly boosted by the early Easter holiday.
The surveys for March follow news earlier this week of a record pounds 1.2bn increase in consumer credit February.
"It's as we thought. Manufacturing is holding up well, and services are just ripping away," said James Barty, an economist at Deutsche Morgan Grenfell.
Even so, almost no City experts think Kenneth Clarke will raise interest rates after what could be his final monetary meeting with Eddie George, Governor of the Bank of England, next week. Any increase in the base rate would almost certainly trigger higher mortgage rates, and the Chancellor is thought unlikely to take that risk so close to the election.
The new purchasing managers' services survey aroused a lot of interest in the City, as the sector is covered by very few statistics despite its importance to the economy. The parallel purchasing managers' index for manufacturing has a strong track record as an early indicator of trends in industry.
The new indicators are harder to interpret, with figures going back only nine months and some uncertainty about the normal seasonal fluctuations.
However, the figures indicated a rapidly rising level of activity, of new business and employment. They also showed that service sector companies are having to pay sharply higher salaries, although not yet passing these on in rising prices.
The activity index jumped from 60.4 in February to 63.3 in March, well above the 50 "boom-bust" watershed. The index for new business rose from 60.9 to 62.4, and for employment from 56.8 to 57.6. Nearly 85 per cent of the companies surveyed expect business to pick up still further this year.
The index of input prices - mainly salaries - was 57, down from 58.9 in February but still indicating a sharp rise. Prices charged have been rising at a gentler pace, with their index at 51.6 in March.
"Inevitably rising costs will find their way out in prices at some point but it depends on how long this very confident mood stays in place," said Peter Thomson, director-general of the Chartered Institute of Purchasing and Supply.
Yesterday's CBI survey showed a modest recovery in the volume of retail sales - the main area of services not covered by the purchasing managers' survey. Kate Barker, the CBI's chief economist, said the improvement was likely to continue into April. The balance reporting higher rather than lower sales was 33 per cent, up from 24 per cent in February.