A Healthy number of trainees qualifying in September will be kept on by the top UK firms.
Of the 14 London and regional firms contacted by The Lawyer, an average of only three trainees per firm were yet to be offered places.
The firms reported increasing workloads as well as better trainee selection procedures.
But Eversheds appeared to be bucking the trend. Among the firms contacted in the survey it will be shedding the largest proportion of trainees in September.
Just under a third - four out of 15 - were not being taken on because they failed to meet the required standards.
A spokesman said it would be recruiting from outside the firm to fill the spaces left by the four trainees.
Leading the poll were Freshfields and Herbert Smith, which offered places to all their current trainees. Allen & Overy said it would have offered places to all trainees but four left voluntarily.
At the height of the recession, firms were regularly letting go around a fifth of their trainees.
The acceptance rate is also high according to the survey results, with an average of only two trainees leaving.
Linklaters reported that two of its trainees had decided to leave law but would be staying on at the firm to work in the personnel department.
Jonathan Bond, graduate recruitment manager at Allen & Overy, said there was greater demand for new solicitors than there had ever been before.
He added that demand for qualifying solicitors was highest in the corporate, property and international capital markets areas.
Jane Bush, partner in charge of recruitment at Dibb Lupton Broomhead, said: "The general feeling is very positive. The market has picked up for trainees and there is a better quality of candidate out there."