Total unemployment fell by 57,000 in the quarter to May to 2.51 million, the lowest since last autumn, while the claimant count was cut by 21,200 in June to 1.48 million, the best figure for more than two years.
Vacancies were up by 24,000 to 529,000, the highest since the autumn of 2008.
But long-term unemployment has reached its highest level since 1996, with 915,000 people out of work for more than a year, an increase of 32,000. Just over 460,000 people have been jobless for more than two years, the highest figure since 1997.
The number of people classed as economically inactive also increased in the latest quarter, up by 87,000 to 9.04 million. The figure included a 44,000 increase in economically inactive students, a 26,000 rise among the long-term sick and 8,000 more people who retired early.
The Office for National Statistics also reported that 29.7 million people were in employment in the three months to May, up 16,000 on the previous quarter, and an increase of 336,000 on a year ago.
Average earnings increased by 1.7% in the year to May, up by 0.2% on the previous month, giving an average weekly wage of £476.
Full-time employment increased by 28,000 to 21.6 million, but the number of part-time workers fell by 12,000 to 8.04 million. Self-employment fell by 28,000 to 4.1 million.
Employment Minister Mark Hoban said: "Our welfare reform agenda is about making a fair and affordable system that incentivises work: giving people the support they need to get off benefits and into a job. Today's figures are encouraging, with the number of people claiming Jobseeker's Allowance down and the number of people in work increasing."