It meant the unemployment rate fell by 0.2% to its lowest since 1975, at 4.5%, the Office for National Statistics (ONS) added.
But wage increases continued to fall further behind inflation.
Excluding bonuses, earnings rose by 2.0% year-on-year. However, inflation had hit an almost four-year high of 2.9% in May.
When the impact of inflation is factored in, real weekly wages fell by 0.5% compared with a year earlier.
Mark Carney: UK wage growth 'anaemic'
UK workers 'frustrated by flatlining pay'
UK inflation rate at near four-year high
"Despite the strong jobs picture... there has been another real-terms fall in total earnings, with the growth in weekly wages low and inflation still rising," said Matt Hughes, senior statistician at the Office for National Statistics.
Those in work climbed to around 32 million, a rise of 324,000 on last year and the largest total since records began in 1971.
The employment rate rose by 0.3% on the quarter to a record high of 74.9%.
"The general picture is little changed on last month, with the overall employment rate and that for women both at record highs, the inactivity rate at a joint record low and the unemployment rate falling to its lowest since early summer 1975," said Mr Hughes.
The unemployment rate for 16 to 24-year-olds was 12.5%, lower than for a year earlier when it was 13.5%, and well below its highest rate of 22.5% in late 2011.
The sluggish pay data may cause Bank of England officials to think again about the need to raise interest rates, after a narrow 5-3 vote by the Monetary Policy Committee (MPC) last month to leave rates at 0.25%.
"The continued weakness of wage growth provides some ammunition to the more dovish members of the MPC that now is not the time to raise interest rates," said Paul Hollingsworth, UK economist at Capital Economics.
He added: "Given the emphasis that some members of the Monetary Policy Committee, including Governor Carney,  have put on wanting to see a clear "firming" in wage growth before they join others in voting to hike interest rates, we still think it is more likely than not that the MPC will hold off for a while longer, rather than raise interest rates imminently."
Meanwhile, Minister for Employment, Damian Hinds said the employment figures were "another reminder that our strong economy is giving record numbers of people the chance to find and stay in work".
But TUC general secretary Frances O'Grady said: "Ministers must set out a plan to get real wages rising across the public and the private sectors."
After trading lower against the dollar, the pound gained ground to trade 0.1% stronger on the day at $1.2858. Sterling also gained 0.1% against the euro, with one pound getting you 1.1217 euros.

LL Camps has operated at Merry Hill in Bushey, Hertfordshire, for the past four years, describing itself as the UK's only American-style summer camp.
A man aged 25, of Borehamwood, had been bailed until September, said police.
They said they would contact families directly should they discover that any children had been affected.
A spokesman for Ofsted said: "We can confirm we have suspended the registrations of these early years settings while investigations are carried out.
"It would be inappropriate to comment further while investigations are ongoing."
In a joint statement, Hertfordshire Police and Hertfordshire County Council said they were working with Ofsted and the "safety and welfare of children is the top priority".
Anyone with specific concerns about a child is asked to call Hertfordshire County Council on 0300 123 4043.
A statement from the LL Camps team was posted its Facebook page on Thursday. It said the camp had closed with "immediate effect" and it "cannot disclose any further information".
The privately-run LL Camps is based at St Margaret's School in Bushey.
A school spokeswoman said LL Camps was "entirely non-related to our school" and St Margaret's had "no links to LL Camps except to hire out a part of the Sports Centre site during the holiday periods".
The LL Camps website, which has been taken offline, stated it had achieved "outstanding early years provider" awards from Ofsted in 2013 and 2014.
It said the camp was about "meeting new friends, expanding interests and learning new skills". It added it was committed to its core values of creating a "caring, nurturing and positive" environment.
UK unemployment fell by 64,000 to 1.49 million in the three months to May, official figures show.
[NEXT_CONCEPT]
A children's summer camp has been closed by Ofsted after a man was arrested on suspicion of possessing indecent images of children.