Demolition firm Brown and Mason will also clear the collapsed section of the boiler house and demolish the rest of the building.
It is set to replace Coleman and Company, which still has responsibility for demolition at the wider site.
Four workers died when the plant collapsed on 23 February.
In a statement, Coleman and Company said it was "hugely disappointing" it had to cease work at the site.
"We all wanted to recover our friends and colleagues and return them to their families," the company said.
"It has been explained to us that this decision has been made for reasons of safe-guarding employee welfare and the preservation of critical evidence."
RWE Npower has announced plans to bring the rest of the building down by a controlled explosive demolition.
The bodies of Christopher Huxtable, 34, from Swansea, Ken Cresswell, 57, and John Shaw, 61, both from Rotherham, have not yet been found following the collapse.
The body of Michael Collings, 53, from Brotton, Teesside, has been recovered from the site.
The cause of the collapse is being investigated jointly by police and The Health and Safety Executive.
A new contractor will take over work within the next month to recover the bodies of three men killed when Didcot A Power Station collapsed.