There has also been an acceleration in the growth of employment in Scotland. The West Midlands was the only part of the country to see faster expansion in job creation last month.
However, the pace of growth in job creation remains below that for output, a finding that was also apparent in a separate UK-wide survey by accountant BDO.
The Bank of Scotland's reading of 58.3 for business output in August is well ahead of the level of 50, which is calculated to separate expansion from contraction.
An acceleration from the 56.7 points registered for July, it is a record reading for the survey which was launched in January 1998.
It is, however, lower than the 60.8 points recorded for the wider UK.
Donald MacRae, chief economist at Bank of Scotland, said: "August's PMI showed the economic recovery continuing in Scotland.
"The latest month's results set two records for the survey showing a sharp increase in business activity and new work accompanied by a strong rise in employment.
"The pace of the economic recovery in Scotland is clearly quickening but would be further enhanced by firms increasing investment."
What might worry some observers who are concerned that recovery is based on an unsustainable consumer boom is that the dominant services sector has continued to expand faster than manufacturing.
Business activity in the services sector saw the most marked month-on-month increase since May 2007, according to Bank of Scotland. More than twice as many firms recorded an increase in business over the previous month as noted a decrease.
This tallies with a separate Barclaycard survey showing that consumer spending rose 4.4% year-on-year in August, the fourth month in a row with growth above 4%. But at 3.4%, spending growth in Scotland lagged the rest of the UK.
Conditions appear tougher in manufacturing, according to the Bank of Scotland survey.
Last month, there was a slight slowing in the pace of growth in the sector which had hit a 29-month high in July. Nevertheless expansion was the second-fastest recorded in the past two-and-a-half years.
As the economy grew, so too did Scotland's jobs market, with employment increasing for the ninth consecutive month, according to Bank of Scotland. The pace of job creation was the fastest since June 2007 and outstripped that for the UK as a whole.
But there are concerns that improving business conditions and confidence are not being matched by a rise in companies' recruitment.
BDO's employment index remained at the same 97.0 level as in July. The index has risen by just 0.6 points in the last five months.
This is consistent with official figures showing the unemployment rate of 7.8% has barely budged in the last year.
Yet its output index, which measures business conditions, rose to 98.3, up from 96.8 the previous month, the highest mark since March 2011.
Neil Craig, partner and head of BDO in Scotland, said: "We're encouraged to see that business conditions and confidence are continuing to improve, and that the UK economy is set for relatively robust expansion for the remainder of 2013.
"However the sting in the tail is that the improvement is not being reflected in businesses' hiring intentions."
He said that with productivity levels marooned at 2005 levels, the pick-up in business output may be partially absorbed by higher levels of output per worker.
The state of the UK labour market will be in focus this week with the publication of labour market data for the three months to the end of July. The City expects to see a slight fall in the rate to 7.7%.
Jobless rates are now under ever-increasing focus because the Bank of England has tied increases in Base Rates to unemployment falling below 7%.