The unemployment rate was 4.9%, down from 5.5% a year earlier but little changed from last month's report, the Office for National Statistics said.
Growth in average weekly earnings including bonuses slowed slightly to an increase of 2.3% on last year.
The proportion of people in work hit a record high of 74.5%.
The ONS said the figures, which only cover one month since the result of the EU referendum, show "continuing improvement" in the jobs market.
But Samuel Tombs, chief UK economist at Pantheon Macroeconomics, warned: "When you scratch beneath the surface, today's labour market figures are not as robust as they first appear."
The rise in people in work "remains supported by surging self-employment", Mr Tombs said.
"The strong growth also reflected a shift towards part-time working; total weekly hours rose by just 0.3% between April and July," he added.
Ben Brettell, senior economist at Hargreaves Lansdown, said the labour market is "still waiting for the Brexit effect", but that it will "gradually become clear over the next few months".
In August the claimant count, including Jobseeker's Allowance, actually went up from the previous month by 2,400 to 771,000.
The claimant count is treated with some caution, though, by economists as the move to Universal Credit has made it much harder to calculate.
One interesting nugget in the employment figures concerns the number of people employed in the public sector.
It is down to 5.33 million, the lowest level since the Office for National Statistics started collecting the figures in 1999 and a drop of 13,000 since March.
The former chancellor, George Osborne, often spoke about the need to "rebalance" the economy.
To an extent, he meant away from financial services and towards manufacturing.
But, more privately, he was actually very comfortable with an economy that was "rebalancing" away from the public sector towards the private sector, where wealth, he would argue, is actually created.
As cuts and wage freezes bite, the public sector has become less attractive as a place to build a career.
Yes, employment is at record levels, but it is the private sector that is on the up.
The unemployment estimates come from the Labour Force Survey in which the ONS talks to 40,000 households every three months.
Although it is a very large survey, there is still a margin of error. The ONS says it is 95% confident that the figure of a 39,000 fall in unemployment is correct to within 78,000.
As the estimated change is smaller than the margin of error, it means the change in unemployment is not statistically significant.
The number of people without jobs and looking for work fell by 39,000 to 1.63 million between May and July, official figures show.