Economists have said the slight fall in UK unemployment to 1.63 million between May and July shows there is yet to be a "Brexit effect" on the jobs market.
The unemployment rate was 4.9%, down from 5.5% a year ago and little changed from last month's rate, Office for National Statistics (ONS) data shows.
Nearly three quarters of people who can work have jobs, a record high rate.
Employment was "resilient" before and after the EU vote, despite predictions of an economic shock, analysts said.
Kallum Pickering, an economist at Berenberg, said: "Although it is still early days, the UK labour market is yet to show any Brexit effect for the period immediately before and after the June 23 vote."
His analysis of the ONS data showed that unemployment fell to 4.7% in July, the first month since the vote.
Ben Brettell, senior economist at Hargreaves Lansdown, said: "The UK's labour market proved resilient in the immediate aftermath of the vote to leave the EU, ONS data has shown.
"This is the latest piece of evidence which shows the economy has fared better than expected since June's referendum."
John Hawksworth, chief economist at PwC, agreed the jobs data showed "no immediate impact from the Brexit vote".
Beneath the surface
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.
Growth in average weekly earnings including bonuses also slowed slightly to an increase of 2.3% on last year. That is still ahead of inflation, which was 0.6% in the year to August.
In signs of a possible slowdown in employment, the claimant count, including Jobseeker's Allowance, went up from the previous month by 2,400 to 771,000.
It comes after a survey this week found employers in six out of nine sectors are less optimistic about adding jobs in the wake of the Brexit vote.
The claimant count is treated with some caution, though, by economists as the move to Universal Credit has made it much harder to calculate.
Analysis: BBC economics editor Kamal Ahmed
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.
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