The report said the seasonally adjusted Scotland Composite Output Index rose again for the fifth month in a row at an above average rate from 50.7 in May to 53.2 in June.
The upturn across Scotland also surpassed that seen at the UK level, placing it third in the UK regional rankings table behind London and the South East.
The growth of new business also accelerated during the month – driven mainly by services firms, according to the bank.
Turning to prices, inflationary pressures showed further signs of easing from the highs seen in the previous two years.
Scotland’s private sector also signalled a fifth monthly rise in new business during June.
The rate of expansion is thought to have quickened mainly due to a faster increase at services providers.
A panel noted that greater demand and a general market improvement helped drive the upturn.
In addition, Scotland recorded the second fastest expansion in new work among the 12 UK areas, behind London.
Much of the Scottish private sector was also found to be largely upbeat at the end of the second quarter.
This sentiment stemmed from planned growth in business, anticipated increases in sales and the launch of new products.
Confidence dipped to a five-month low, however, running below the long-run average.
Judith Cruickshank, chair of the Scotland Board of Royal Bank of Scotland, commented: “The Scottish private sector signalled a stronger performance midway through the year.
“The upturn was largely supported by a quicker expansion across the services sector, while manufacturing continued to exhibit weakness despite registering a slight increase in production.
“The diverging trends between the two sectors are a concern as dependence on services grows.
“This is highlighted by a quicker expansion in new business at services firms, while manufacturers signalled a third monthly contraction in factory orders.
“New orders also give an indication of business activity in the coming months, and the data from June signals growth will remain skewed towards services.
“Furthermore, outlook expectations across the two sectors also showed more subdued sentiment across manufacturers, while service providers remained upbeat in comparison.
“In terms of prices, inflationary pressures eased in June, with cost burdens rising at the softest rate since May 2021. Panellists largely attributed the upturn in overall cost burdens to increasing labour costs.”