Scotland's chief statistician estimated services grew by 0.5% and production by 0.3% between April and June, while construction contracted by 1.9%.
UK output as a whole grew by 0.7% over the same period.
Over the past year, the Scottish economy grew by 0.7% - a third of the UK rate of 2.1%.
In the first three months of the year, there was no growth in Scotland.
Scottish GDP per person - which takes population changes into account - grew by 0.3% during the second quarter, compared with 0.4% for the UK.
The report indicated that growth in Scottish GDP over the past year was driven by growth in the services industry, particularly in business services and finance.
However, that was "tempered" by contractions in the construction and production industries, especially electricity and gas, following the closure in March of Scotland's last coal-fired power station.
It was estimated that the closure resulted in a reduction of Scottish GDP of about 0.2 percentage points in the second quarter.
The economic report added: "As this was a one-off closure it will not have an ongoing impact on the growth of the Scottish economy."
Reacting to the figures, Scottish Chambers of Commerce chief executive Liz Cameron said it was "good news" that Scotland's economic growth rate had increased but added that there was "still a great deal of work to be done".
She said: "To put this in perspective, the Scottish economy has grown in a year at almost the same rate that the UK economy has grown in just three months.
"These figures underline the fact that Scotland's economic performance has been significantly lower than that of the UK as a whole for a full year and, whilst we are now seeing welcome growth in our production and service sectors, construction has been contracting at a significant rate for two consecutive quarters."
Colin Borland, head of external affairs in Scotland for the Federation of Small Businesses, said: "These pre-referendum statistics might feel like a history lesson, but they teach us that Scottish growth was weak even before June's historic vote.
"Scotland needs to strive for growth levels at least as good as the UK average.
"We look forward to the SNP talking business when they meet in Glasgow this week.
"In addition, the UK government needs to put the welfare of the economy at the top of its priority list as it formulates its approach to leaving the EU.
"The litmus test for every Brexit policy must be the impact on our high streets, small businesses and local communities."
Output from the Scottish economy grew by 0.4% in the second quarter of this year but lagged behind the UK as a whole, according to official figures.