The squeeze on real incomes continues to grow, though at a slower pace.
Average weekly earnings increased by 2.1% compared with a year earlier - slightly higher than last month's 2% increase.
But with inflation standing at 2.6%, real earnings still fell by 0.5%, the ONS figures showed.
At 75.1%, the proportion of people in work is the highest it has been since 1971 - partly due to the introduction of a later state pension age for women.
There were 32.07 million people in work in the three months to June - 338,000 more than for the same period last year.
"The employment picture remains strong, with a new record high employment rate and another fall in the unemployment rate. Despite the strong jobs picture, however, real earnings continue to decline," said Office for National Statistics senior labour market statistician Matt Hughes.
Analysis
Andy Verity, economics correspondent
Pay rises improved, up by 2.1% (excluding bonuses) compared with a consensus prediction of 2%. Maybe the economic theory was right after all - and pay is now ticking up because labour markets are tight.
It remains, however, a long way short of what would be required to trigger the sort of wage-price spiral about which central bankers have been hyper-vigilant since the 1970s.
That employees are prepared to accept wages that shrink by a tiny bit less than they did the last time these figures came out does not exactly bespeak a dramatic new assertion of workers' bargaining power.
Read more from Andy here
Jobs were created in the construction, accommodation and food services sectors and transport and storage industries.
Pay rises improved, up by 2.1% (excluding bonuses) compared with a consensus prediction of 2%.
The increase prompted some economists to suggest that wages may finally be responding to an economy which is closing in on full employment.
It remains, however, a long way short of what would be required to trigger the sort of wage-price spiral about which central bankers have been hyper-vigilant since the 1970s.
That employees are prepared to accept wages that shrink by a tiny bit less than they did the last time these figures came out does not exactly bespeak a dramatic new assertion of workers' bargaining power.
Ruth Gregory, UK economist at Capital Economics said the figures gave some signs that the tighter labour market was leading to a recovery in wage growth.
"Inflation is likely to fall back next year as the impact of the drop in the pound dwindles. What's more, the tightness of the labour market should deliver further rises in nominal wage growth over the coming quarters," she said.
The pound rose rose against both the dollar and the euro following the positive news on jobs, recovering some of the ground lost on Tuesday, before losing most of those gains.
However, productivity - or output per worker - continued to decline, the ONS said, issuing preliminary figures for the second quarter. Productivity was 0.1% lower than in the first quarter and "remains at around the same level as its pre-downturn peak".
The number of non-UK nationals employed in the UK workforce continued to increase, rising 109,000 to 3.56 million compared to a year earlier.
Within that non-UK nationals from in the EU continued to rise, while workers from outside the EU decreased by 18,000 from a year ago.
The number of people on zero hours contracts as their main job fell 20,000 compared to a year earlier to 883,000 people.
Enter your details below. Source: Office for National Statistics.

How to pronounce the budget retailer's name has caused a basket-load of debate.
Some claim "Pry-mark" is an upmarket affectation. Others say "Pree-mark" is a northern thing.
But Primark have tried to settle the matter once and for all, writing on their website: "We like to use Pr-Eye-Mark."
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But Primark isn't the only brand that people mispronounce...
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Unemployment in the UK fell by 57,000 in the three months to June, official figures show, bringing the jobless rate down to 4.4% - its lowest since 1975.
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Pry-mark or Pree-mark?