Unemployment fell by 27,000 in the three months to February to 1.34 million, official Office for National Statistics (ONS) figures show.
The number of people in work was also virtually unchanged at a record high of 32.7 million, with a jump of 179,000.
The figure has increased by 457,000 over the past year, all among full-time employees and the self-employed.
Average weekly earnings, excluding bonuses, had an estimated rise of 3.4%, before adjusting for inflation.
When adjusted for inflation, pay, including bonuses, increased by 1.5% on the year, the highest figure since the summer of 2016.
The UK’s unemployment rate of 3.9% is now lower than at any time since the end of 1975.
ONS deputy head of labour market statistics Matt Hughes said: “The jobs market remains robust, with the number of people in work continuing to grow.
“The increase over the past year is all coming from full-timers, both employees and the self-employed.
“Earnings have now been growing ahead of inflation for over a year, but in real terms, wage levels have not yet returned to their pre-downturn peak.”
Employment Minister Alok Sharma said: “The UK jobs market continues to go from strength to strength, proving the underlying resilience of the British economy.
“But we must not take this for granted. We need to work urgently to get behind a Brexit deal that protects this jobs record and gives employers the certainty to continue to invest in their workforce and boost wages.”
Mike Amesbury, Labour’s shadow employment minister, said: “Behind today’s headline figures, average wages are still less than they were 10 years ago and in-work poverty is rising faster than employment.
“Too many people are trapped in low-paid, insecure work and 70% of children in poverty now live in working families.”
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The number of economically inactive people fell by 114,000 in the latest quarter to 8.53 million, a rate of just under 21%, the joint lowest on record.
The number of vacancies was almost unchanged at 852,000.
Anxiety over Brexit has deterred some businesses from investing – but not, it would appear, hiring more workers as yet.
Hiring plans tend to lag behind changes in economic activity, as employers wait to assess changes in demand, so the resilience of the labour market is perhaps unsurprising – particularly as consumer spending remains solid for now.
But economists say employment could yet falter later in the year if the uncertainty is drawn out.
Wages growth continues to comfortably outpace inflation compared with a year ago (although in real terms, the level of average wages remains below the pre-crisis levels).
The level of vacancies is down on the record of 864,000 seen at the start of the year, in another sign of strong demand.
A note of caution when quoting these: the ONS recently overhauled the way the numbers are counted – because as they are based on a sample of firms, they stress this is an estimate and perhaps “not precise”.
TUC general secretary Frances O’Grady said: “This modest pay growth is doing little for workers still feeling the effects of the longest pay squeeze for 200 years.
“And with over half of those in poverty living in working households, we need a more ambitious plan to support jobs and wages.”
Federation of Small Businesses national chairman Mike Cherry said: “At a time when political uncertainty is making it impossible to plan and operating costs are spiralling, a tight labour market represents yet another headache for small business owners.
“One in five small UK employers rely on staff from the EU. The sharp drop in European arrivals is a real concern for many smaller firms, particularly those in sectors such as construction, care and engineering, where the contribution of EU team members is so vital. One in three small firms now say lack of access to the right personnel is a major barrier to growth.”
Thomas Pugh, UK economist at Capital Economics, said: “We suspect that this could mark the peak of employment growth, as the Brexit uncertainty reached its crescendo and the surveys turned down sharply in March.”
He added that employment growing more slowly than output could ease some pressure on labour costs and said that he did not expect any interest rate rise until the second half of 2020.