Demand is also putting pressure on the cost of manufacturers' inputs and raw materials. Many firms have reported rising prices for commodities, plastic, paper and steel products as shortages develop.
The CIPS Purchasing Managers' Index shows factory output has been expanding consistently for two years, but the rate of increase is now at its lowest this year.
The combination of slower growth and upward pressure on prices left City economists divided on the message for interest rates. Most expect the Chancellor and the Governor of the Bank of England to agree to keep rates steady when they meet next week.
The PMI fell from 62 to 59.6 in August, adjusting for normal seasonal effects. Nearly a third of firms had reported difficulty in obtaining supplies, with delivery times reaching a record high. New orders were boosted by demand from Europe and America.
Employment grew for the 11th month running - a finding at odds with official figures showing continued job-shedding.
'This strong recovery presents a great opportunity for British manufacturers,' said Peter Thomson, director-general of CIPS. 'However, they urgently need to increase capacity and shorten delivery times to customers if they are to take full advantage of the growing demand.'
Simon Briscoe, economist at Warburg Securities, said the fall in the growth rate of factory output since June would reduce pressure to raise interest rates.
The Chicago Purchasing Managers' Index sent a similar message about the US recovery, with the prices component surging but activity relatively subdued.
Factory orders in the US also fell for the first time in five months, by 2.3 per cent between June and July, calming fears of an early rise in interest rates and undermining the dollar against the mark on the foreign exchanges.
In France, the seasonally adjusted number of unemployed fell by 10,800 in July to 3,322,800.
The jobless rate based on International Labour Organisation criteria, which exclude jobseekers who did any work last month, was unchanged at 12.6 per cent, in line with economists' forecasts.
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