According to figures from the Office for National Statistics, the number of people claiming Jobseeker's Allowance fell in June by 20,800 to 1.46 million.
The number of people in work rose by 160,000 in the three months to May, which marked the biggest rise since August 2006.
Lynne Hardman, managing director at recruitment consultancy Badenoch & Clark, said: "It is clearly a positive sign to see unemployment come down at a rate of 7.9%, particularly as this is the fifth month in a row and reflects the general improved health of the job market.
"Over the past few months we have seen a steady increase in recruitment activity across various sectors but employers remain cautious in their approach to recruitment. Equally, there are still people in part-time or temporary positions who would prefer to be working permanently or full-time. There are also cases of people taking positions for which they are over-qualified.
"We remain concerned that the impact of the Emergency Budget and the promised public-sector cuts is yet to be felt. Only time will tell whether private-sector growth will be able to absorb the job losses expected in the public sector."
Nigel Meager, director of the Institute for Employment Studies, added: "There are some encouraging signs of stronger labour market demand in this month’s figures, with unemployment down by 34,000, and employment growing by 160,000. Vacancies have also risen again, though they remain well below pre-recession levels. However, it is of serious concern that long-term unemployment has increased again to over three-quarters of a million. Urgent policy action is needed to support this group back into work to avoid the damaging long-term social impacts experienced in previous recessions."
"Official figures are yet to register a major downturn in public sector employment, but this will inevitably follow. Recruitment to public administration has dropped dramatically, with less than one vacancy per 100 employees, far lower than any other sector apart from construction, and half the level of a year ago. This is also the only sector in which redundancies have increased over the last year. These trends will accelerate in the months ahead, and it is unlikely that the real, but muted recovery in the private sector will be sufficient to offset significant public sector job loss."